Sec 2 G1 Maths Tutor Bukit Timah

Did you know? A linear function can be treated like a little machine: choose an input, apply two simple steps and read the output. One rule might multiply the input by three and add two; another might multiply it by negative two and add seven. A table turns those instructions into visible relationships long before a student draws a straight line.

Quick answer: read the function table before the graph

For the rule y=3x+2, input x goes into the first column and output y into the second. At x=0, y=2; at x=1, y=5; at x=2, y=8. The ordered pairs are (0,2), (1,5) and (2,8). The gradient is three, meaning y increases by three for each one-unit increase in x. The y-intercept is two, the output at x=0. This is why a table can help verify the graph, and a graph can help interpret the table.

This article is a quick-learning guide for families seeking a Secondary 2 G1 Mathematics tutor in Bukit Timah. The MOE G1 Secondary Two outline explicitly includes Cartesian coordinates, plotting ordered pairs, linear functions y=ax+b and graphs with positive or negative gradient. This guide concentrates on input–output table reasoning, which differs from our separate guides devoted to plotting across quadrants or computing a gradient from two given points.

A 60-second diagnostic using one rule

Ask the student to use y=2x−3 with x=−1,0,2. The outputs are −5, −3, 1. Then ask which x gives y=5: solving 5=2x−3 gives x=4. Finally ask whether the graph rises or falls to the right: it rises, because the coefficient of x is +2. Different errors point to different gaps—signed substitution, reverse equations or understanding the gradient.

Why a function machine helps

The x-column is not merely the left side of a table

A table’s input labels describe possible independent-variable values. The output column records the result of applying the rule to each input. In y=4x+1, the machine multiplies by four, then adds one. For x=2 it produces nine. Reversing the columns would assert that y is the input; the mathematical relationship could be rewritten, but the requested pairs and graph labels would be different.

The constant term is the starting output

When the rule is y=4x+1, putting x=0 removes the x-term and reveals y=1. That is the y-intercept, written as the point (0,1). In a fixed-fee story the intercept may represent an initial charge; in a temperature story it may represent a starting reading. But not every line represents a real-world process—always check the specified units and permitted inputs before translating an intercept into a factual claim.

The same change repeats across equal input steps

For y=−2x+5, increasing x from zero to one changes y from five to three. Increasing x from one to two changes y from three to one. The rate is negative two per input unit, which is the gradient of the line. If a table jumps from x=0 to x=3, the change in y is three times the gradient, not just the gradient itself. Divide change in y by change in x to check the claimed slope.

Direct proportion is a special case. The line y=3x is directly proportional because its ratio y/x is constant for positive nonzero x and it passes through origin. But y=3x+5 does not: the five is a fixed intercept. It remains a linear function, yet doubling x does not in general double y. This distinction is an important bridge between ratio work and graphs.

Find the input when an output is given

A question may ask, ‘For y=3x+2, which input gives output seventeen?’ Write 17=3x+2, subtract two and divide by three to get x=5. This is an equation, not a table lookup requiring every intermediate number to be listed. The quick check is to feed the recovered input five back into the original rule. In a context, label the input as visits, hours or items, as appropriate.

98 original linear function and table clinics

The examples begin with straightforward input–output tables and reverse-input questions, then compare vertical shifts and use hypothetical applications. These are original worked teaching cases rather than copied examinations. Choose a few at a time and ask the pupil to explain why the rate stays constant across rows. A correctly computed output must still be placed under the correct input label.

1. Function-table clinic 1: y=x+3

Question. Using y=x+3, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 1, then adds the signed constant 3. At x=-2, y=1(-2)+(3)=1. At x=0, y=3. At x=3, y=6. The ordered pairs are (-2,1), (0,3), (3,6).

The first error to diagnose. A child might put output 1 into the x-column, turning the pair into (1,-2). Another might omit brackets for negative x, writing 1−2 as subtraction rather than 1×(−2).

Verify from the rule. The output change between x=-2 and x=0 is 2 over 2 input units, a rate of 1 per input unit. Between x=0 and x=3, the change is 3 over 3 input units, the same slope 1. A linear function has constant rate of change.

Independent transfer. Hide the table and calculate y at x=4, which should be 7. Explain whether y rises or falls as x grows: the sign of the coefficient 1 answers that question before a calculator is used.

2. Reverse function clinic 1: find the input

Question. A function follows y=x+3. Which x-value produces output y=-1?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: -1=1x+(3). Subtract the constant 3 on both sides to obtain -4=1x. Divide by 1, yielding x=-4. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=-1 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=-4 into the original function: 1(-4)+(3)=-1. The output matches exactly, so (-4,-1) lies on the line.

Independent transfer. Now change the target output to 0. The required x becomes -3 because one extra x-unit changes y by the fixed slope 1. Explain this without repeating a full equation solve.

3. Compare two linear rules 1: fixed output shift

Question. Compare y=x+3 with y=x+5. What changes at input x=-1, and do the graphs have the same gradient?

Did you know? A function turns an input into one output under a specified rule. At x=-1, the original gives 2 and the new rule gives 4. Every output of the second is 2 higher for the same input because only the constant term changed. Both rules have gradient 1; their y-intercepts are (0,3) and (0,5).

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 5 and 7, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 2 but unchanged intercept 3. Explain how that changes the rate per input rather than moving every output by a fixed amount.

4. Function-table clinic 2: y=2x+1

Question. Using y=2x+1, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 2, then adds the signed constant 1. At x=-2, y=2(-2)+(1)=-3. At x=0, y=1. At x=3, y=7. The ordered pairs are (-2,-3), (0,1), (3,7).

The first error to diagnose. A child might put output -3 into the x-column, turning the pair into (-3,-2). Another might omit brackets for negative x, writing 2−2 as subtraction rather than 2×(−2).

Verify from the rule. The output change between x=-2 and x=0 is 4 over 2 input units, a rate of 2 per input unit. Between x=0 and x=3, the change is 6 over 3 input units, the same slope 2. A linear function has constant rate of change.

Independent transfer. Hide the table and calculate y at x=4, which should be 9. Explain whether y rises or falls as x grows: the sign of the coefficient 2 answers that question before a calculator is used.

5. Reverse function clinic 2: find the input

Question. A function follows y=2x+1. Which x-value produces output y=-1?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: -1=2x+(1). Subtract the constant 1 on both sides to obtain -2=2x. Divide by 2, yielding x=-1. This reverses the order of the function-machine steps.

Verify from the rule. Put x=-1 into the original function: 2(-1)+(1)=-1. The output matches exactly, so (-1,-1) lies on the line.

Independent transfer. Now change the target output to 1. The required x becomes 0 because one extra x-unit changes y by the fixed slope 2. Explain this without repeating a full equation solve.

6. Compare two linear rules 2: fixed output shift

Question. Compare y=2x+1 with y=2x+3. What changes at input x=-1, and do the graphs have the same gradient?

Did you know? A function turns an input into one output under a specified rule. At x=-1, the original gives -1 and the new rule gives 1. Every output of the second is 2 higher for the same input because only the constant term changed. Both rules have gradient 2; their y-intercepts are (0,1) and (0,3).

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 5 and 7, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 3 but unchanged intercept 1. Explain how that changes the rate per input rather than moving every output by a fixed amount.

7. Function-table clinic 3: y=3x−2

Question. Using y=3x−2, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 3, then adds the signed constant -2. At x=-2, y=3(-2)+(-2)=-8. At x=0, y=-2. At x=3, y=7. The ordered pairs are (-2,-8), (0,-2), (3,7).

The first error to diagnose. A child might put output -8 into the x-column, turning the pair into (-8,-2). Another might omit brackets for negative x, writing 3−2 as subtraction rather than 3×(−2).

Verify from the rule. The output change between x=-2 and x=0 is 6 over 2 input units, a rate of 3 per input unit. Between x=0 and x=3, the change is 9 over 3 input units, the same slope 3. A linear function has constant rate of change.

Independent transfer. Hide the table and calculate y at x=4, which should be 10. Explain whether y rises or falls as x grows: the sign of the coefficient 3 answers that question before a calculator is used.

8. Reverse function clinic 3: find the input

Question. A function follows y=3x−2. Which x-value produces output y=4?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 4=3x+(-2). Subtract the constant -2 on both sides to obtain 6=3x. Divide by 3, yielding x=2. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=4 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=2 into the original function: 3(2)+(-2)=4. The output matches exactly, so (2,4) lies on the line.

Independent transfer. Now change the target output to 7. The required x becomes 3 because one extra x-unit changes y by the fixed slope 3. Explain this without repeating a full equation solve.

9. Compare two linear rules 3: fixed output shift

Question. Compare y=3x−2 with y=3x. What changes at input x=-1, and do the graphs have the same gradient?

Did you know? A function turns an input into one output under a specified rule. At x=-1, the original gives -5 and the new rule gives -3. Every output of the second is 2 higher for the same input because only the constant term changed. Both rules have gradient 3; their y-intercepts are (0,-2) and (0,0).

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 4 and 6, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 4 but unchanged intercept -2. Explain how that changes the rate per input rather than moving every output by a fixed amount.

10. Function-table clinic 4: y=−x+4

Question. Using y=−x+4, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by -1, then adds the signed constant 4. At x=-2, y=-1(-2)+(4)=6. At x=0, y=4. At x=3, y=1. The ordered pairs are (-2,6), (0,4), (3,1).

The first error to diagnose. A child might put output 6 into the x-column, turning the pair into (6,-2). Another might omit brackets for negative x, writing -1−2 as subtraction rather than -1×(−2).

Verify from the rule. The output change between x=-2 and x=0 is -2 over 2 input units, a rate of -1 per input unit. Between x=0 and x=3, the change is -3 over 3 input units, the same slope -1. A linear function has constant rate of change.

Independent transfer. Hide the table and calculate y at x=4, which should be 0. Explain whether y rises or falls as x grows: the sign of the coefficient -1 answers that question before a calculator is used.

11. Reverse function clinic 4: find the input

Question. A function follows y=−x+4. Which x-value produces output y=8?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 8=-1x+(4). Subtract the constant 4 on both sides to obtain 4=-1x. Divide by -1, yielding x=-4. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=8 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=-4 into the original function: -1(-4)+(4)=8. The output matches exactly, so (-4,8) lies on the line.

Independent transfer. Now change the target output to 7. The required x becomes -3 because one extra x-unit changes y by the fixed slope -1. Explain this without repeating a full equation solve.

12. Compare two linear rules 4: fixed output shift

Question. Compare y=−x+4 with y=−x+6. What changes at input x=-1, and do the graphs have the same gradient?

Did you know? A function turns an input into one output under a specified rule. At x=-1, the original gives 5 and the new rule gives 7. Every output of the second is 2 higher for the same input because only the constant term changed. Both rules have gradient -1; their y-intercepts are (0,4) and (0,6).

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 2 and 4, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 0 but unchanged intercept 4. Explain how that changes the rate per input rather than moving every output by a fixed amount.

13. Function-table clinic 5: y=-2x+7

Question. Using y=-2x+7, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by -2, then adds the signed constant 7. At x=-2, y=-2(-2)+(7)=11. At x=0, y=7. At x=3, y=1. The ordered pairs are (-2,11), (0,7), (3,1).

The first error to diagnose. A child might put output 11 into the x-column, turning the pair into (11,-2). Another might omit brackets for negative x, writing -2−2 as subtraction rather than -2×(−2).

Verify from the rule. The output change between x=-2 and x=0 is -4 over 2 input units, a rate of -2 per input unit. Between x=0 and x=3, the change is -6 over 3 input units, the same slope -2. A linear function has constant rate of change.

Independent transfer. Hide the table and calculate y at x=4, which should be -1. Explain whether y rises or falls as x grows: the sign of the coefficient -2 answers that question before a calculator is used.

14. Reverse function clinic 5: find the input

Question. A function follows y=-2x+7. Which x-value produces output y=9?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 9=-2x+(7). Subtract the constant 7 on both sides to obtain 2=-2x. Divide by -2, yielding x=-1. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=9 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=-1 into the original function: -2(-1)+(7)=9. The output matches exactly, so (-1,9) lies on the line.

Independent transfer. Now change the target output to 7. The required x becomes 0 because one extra x-unit changes y by the fixed slope -2. Explain this without repeating a full equation solve.

15. Compare two linear rules 5: fixed output shift

Question. Compare y=-2x+7 with y=-2x+9. What changes at input x=-1, and do the graphs have the same gradient?

Did you know? A function turns an input into one output under a specified rule. At x=-1, the original gives 9 and the new rule gives 11. Every output of the second is 2 higher for the same input because only the constant term changed. Both rules have gradient -2; their y-intercepts are (0,7) and (0,9).

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 3 and 5, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient -1 but unchanged intercept 7. Explain how that changes the rate per input rather than moving every output by a fixed amount.

16. Function-table clinic 6: y=4x−5

Question. Using y=4x−5, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 4, then adds the signed constant -5. At x=-2, y=4(-2)+(-5)=-13. At x=0, y=-5. At x=3, y=7. The ordered pairs are (-2,-13), (0,-5), (3,7).

The first error to diagnose. A child might put output -13 into the x-column, turning the pair into (-13,-2). Another might omit brackets for negative x, writing 4−2 as subtraction rather than 4×(−2).

Verify from the rule. The output change between x=-2 and x=0 is 8 over 2 input units, a rate of 4 per input unit. Between x=0 and x=3, the change is 12 over 3 input units, the same slope 4. A linear function has constant rate of change.

Independent transfer. Hide the table and calculate y at x=4, which should be 11. Explain whether y rises or falls as x grows: the sign of the coefficient 4 answers that question before a calculator is used.

17. Reverse function clinic 6: find the input

Question. A function follows y=4x−5. Which x-value produces output y=3?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 3=4x+(-5). Subtract the constant -5 on both sides to obtain 8=4x. Divide by 4, yielding x=2. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=3 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=2 into the original function: 4(2)+(-5)=3. The output matches exactly, so (2,3) lies on the line.

Independent transfer. Now change the target output to 7. The required x becomes 3 because one extra x-unit changes y by the fixed slope 4. Explain this without repeating a full equation solve.

18. Compare two linear rules 6: fixed output shift

Question. Compare y=4x−5 with y=4x−3. What changes at input x=-1, and do the graphs have the same gradient?

Did you know? A function turns an input into one output under a specified rule. At x=-1, the original gives -9 and the new rule gives -7. Every output of the second is 2 higher for the same input because only the constant term changed. Both rules have gradient 4; their y-intercepts are (0,-5) and (0,-3).

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 3 and 5, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 5 but unchanged intercept -5. Explain how that changes the rate per input rather than moving every output by a fixed amount.

19. Function-table clinic 7: y=-3x+2

Question. Using y=-3x+2, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by -3, then adds the signed constant 2. At x=-2, y=-3(-2)+(2)=8. At x=0, y=2. At x=3, y=-7. The ordered pairs are (-2,8), (0,2), (3,-7).

The first error to diagnose. A child might put output 8 into the x-column, turning the pair into (8,-2). Another might omit brackets for negative x, writing -3−2 as subtraction rather than -3×(−2).

Verify from the rule. The output change between x=-2 and x=0 is -6 over 2 input units, a rate of -3 per input unit. Between x=0 and x=3, the change is -9 over 3 input units, the same slope -3. A linear function has constant rate of change.

Independent transfer. Hide the table and calculate y at x=4, which should be -10. Explain whether y rises or falls as x grows: the sign of the coefficient -3 answers that question before a calculator is used.

20. Reverse function clinic 7: find the input

Question. A function follows y=-3x+2. Which x-value produces output y=14?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 14=-3x+(2). Subtract the constant 2 on both sides to obtain 12=-3x. Divide by -3, yielding x=-4. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=14 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=-4 into the original function: -3(-4)+(2)=14. The output matches exactly, so (-4,14) lies on the line.

Independent transfer. Now change the target output to 11. The required x becomes -3 because one extra x-unit changes y by the fixed slope -3. Explain this without repeating a full equation solve.

21. Compare two linear rules 7: fixed output shift

Question. Compare y=-3x+2 with y=-3x+4. What changes at input x=-1, and do the graphs have the same gradient?

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are -4 and -2, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient -2 but unchanged intercept 2. Explain how that changes the rate per input rather than moving every output by a fixed amount.

22. Function-table clinic 8: y=5x

Question. Using y=5x, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 5, then adds the signed constant 0. At x=-2, y=5(-2)+(0)=-10. At x=0, y=0. At x=3, y=15. The ordered pairs are (-2,-10), (0,0), (3,15).

The first error to diagnose. A child might put output -10 into the x-column, turning the pair into (-10,-2). Another might omit brackets for negative x, writing 5−2 as subtraction rather than 5×(−2).

Verify from the rule. The output change between x=-2 and x=0 is 10 over 2 input units, a rate of 5 per input unit. Between x=0 and x=3, the change is 15 over 3 input units, the same slope 5. A linear function has constant rate of change.

Independent transfer. Hide the table and calculate y at x=4, which should be 20. Explain whether y rises or falls as x grows: the sign of the coefficient 5 answers that question before a calculator is used.

23. Reverse function clinic 8: find the input

Question. A function follows y=5x. Which x-value produces output y=-5?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: -5=5x+(0). Subtract the constant 0 on both sides to obtain -5=5x. Divide by 5, yielding x=-1. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=-5 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=-1 into the original function: 5(-1)+(0)=-5. The output matches exactly, so (-1,-5) lies on the line.

Independent transfer. Now change the target output to 0. The required x becomes 0 because one extra x-unit changes y by the fixed slope 5. Explain this without repeating a full equation solve.

24. Compare two linear rules 8: fixed output shift

Question. Compare y=5x with y=5x+2. What changes at input x=-1, and do the graphs have the same gradient?

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 10 and 12, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 6 but unchanged intercept 0. Explain how that changes the rate per input rather than moving every output by a fixed amount.

25. Function-table clinic 9: y=2x−7

Question. Using y=2x−7, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 2, then adds the signed constant -7. At x=-2, y=2(-2)+(-7)=-11. At x=0, y=-7. At x=3, y=-1. The ordered pairs are (-2,-11), (0,-7), (3,-1).

The first error to diagnose. A child might put output -11 into the x-column, turning the pair into (-11,-2). Another might omit brackets for negative x, writing 2−2 as subtraction rather than 2×(−2).

Independent transfer. Hide the table and calculate y at x=4, which should be 1. Explain whether y rises or falls as x grows: the sign of the coefficient 2 answers that question before a calculator is used.

26. Reverse function clinic 9: find the input

Question. A function follows y=2x−7. Which x-value produces output y=-3?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: -3=2x+(-7). Subtract the constant -7 on both sides to obtain 4=2x. Divide by 2, yielding x=2. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=-3 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=2 into the original function: 2(2)+(-7)=-3. The output matches exactly, so (2,-3) lies on the line.

Independent transfer. Now change the target output to -1. The required x becomes 3 because one extra x-unit changes y by the fixed slope 2. Explain this without repeating a full equation solve.

27. Compare two linear rules 9: fixed output shift

Question. Compare y=2x−7 with y=2x−5. What changes at input x=-1, and do the graphs have the same gradient?

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are -3 and -1, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 3 but unchanged intercept -7. Explain how that changes the rate per input rather than moving every output by a fixed amount.

28. Function-table clinic 10: y=-2x−3

Question. Using y=-2x−3, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by -2, then adds the signed constant -3. At x=-2, y=-2(-2)+(-3)=1. At x=0, y=-3. At x=3, y=-9. The ordered pairs are (-2,1), (0,-3), (3,-9).

The first error to diagnose. A child might put output 1 into the x-column, turning the pair into (1,-2). Another might omit brackets for negative x, writing -2−2 as subtraction rather than -2×(−2).

Independent transfer. Hide the table and calculate y at x=4, which should be -11. Explain whether y rises or falls as x grows: the sign of the coefficient -2 answers that question before a calculator is used.

29. Reverse function clinic 10: find the input

Question. A function follows y=-2x−3. Which x-value produces output y=5?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 5=-2x+(-3). Subtract the constant -3 on both sides to obtain 8=-2x. Divide by -2, yielding x=-4. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=5 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=-4 into the original function: -2(-4)+(-3)=5. The output matches exactly, so (-4,5) lies on the line.

Independent transfer. Now change the target output to 3. The required x becomes -3 because one extra x-unit changes y by the fixed slope -2. Explain this without repeating a full equation solve.

30. Compare two linear rules 10: fixed output shift

Question. Compare y=-2x−3 with y=-2x−1. What changes at input x=-1, and do the graphs have the same gradient?

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are -7 and -5, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient -1 but unchanged intercept -3. Explain how that changes the rate per input rather than moving every output by a fixed amount.

31. Function-table clinic 11: y=x−6

Question. Using y=x−6, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 1, then adds the signed constant -6. At x=-2, y=1(-2)+(-6)=-8. At x=0, y=-6. At x=3, y=-3. The ordered pairs are (-2,-8), (0,-6), (3,-3).

The first error to diagnose. A child might put output -8 into the x-column, turning the pair into (-8,-2). Another might omit brackets for negative x, writing 1−2 as subtraction rather than 1×(−2).

Independent transfer. Hide the table and calculate y at x=4, which should be -2. Explain whether y rises or falls as x grows: the sign of the coefficient 1 answers that question before a calculator is used.

32. Reverse function clinic 11: find the input

Question. A function follows y=x−6. Which x-value produces output y=-7?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: -7=1x+(-6). Subtract the constant -6 on both sides to obtain -1=1x. Divide by 1, yielding x=-1. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=-7 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=-1 into the original function: 1(-1)+(-6)=-7. The output matches exactly, so (-1,-7) lies on the line.

Independent transfer. Now change the target output to -6. The required x becomes 0 because one extra x-unit changes y by the fixed slope 1. Explain this without repeating a full equation solve.

33. Compare two linear rules 11: fixed output shift

Question. Compare y=x−6 with y=x−4. What changes at input x=-1, and do the graphs have the same gradient?

Did you know? A function turns an input into one output under a specified rule. At x=-1, the original gives -7 and the new rule gives -5. Every output of the second is 2 higher for the same input because only the constant term changed. Both rules have gradient 1; their y-intercepts are (0,-6) and (0,-4).

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are -4 and -2, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 2 but unchanged intercept -6. Explain how that changes the rate per input rather than moving every output by a fixed amount.

34. Function-table clinic 12: y=−x−2

Question. Using y=−x−2, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by -1, then adds the signed constant -2. At x=-2, y=-1(-2)+(-2)=0. At x=0, y=-2. At x=3, y=-5. The ordered pairs are (-2,0), (0,-2), (3,-5).

The first error to diagnose. A child might put output 0 into the x-column, turning the pair into (0,-2). Another might omit brackets for negative x, writing -1−2 as subtraction rather than -1×(−2).

Independent transfer. Hide the table and calculate y at x=4, which should be -6. Explain whether y rises or falls as x grows: the sign of the coefficient -1 answers that question before a calculator is used.

35. Reverse function clinic 12: find the input

Question. A function follows y=−x−2. Which x-value produces output y=-4?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: -4=-1x+(-2). Subtract the constant -2 on both sides to obtain -2=-1x. Divide by -1, yielding x=2. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=-4 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=2 into the original function: -1(2)+(-2)=-4. The output matches exactly, so (2,-4) lies on the line.

Independent transfer. Now change the target output to -5. The required x becomes 3 because one extra x-unit changes y by the fixed slope -1. Explain this without repeating a full equation solve.

36. Compare two linear rules 12: fixed output shift

Question. Compare y=−x−2 with y=−x. What changes at input x=-1, and do the graphs have the same gradient?

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are -4 and -2, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 0 but unchanged intercept -2. Explain how that changes the rate per input rather than moving every output by a fixed amount.

37. Function-table clinic 13: y=3x+5

Question. Using y=3x+5, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 3, then adds the signed constant 5. At x=-2, y=3(-2)+(5)=-1. At x=0, y=5. At x=3, y=14. The ordered pairs are (-2,-1), (0,5), (3,14).

The first error to diagnose. A child might put output -1 into the x-column, turning the pair into (-1,-2). Another might omit brackets for negative x, writing 3−2 as subtraction rather than 3×(−2).

Independent transfer. Hide the table and calculate y at x=4, which should be 17. Explain whether y rises or falls as x grows: the sign of the coefficient 3 answers that question before a calculator is used.

38. Reverse function clinic 13: find the input

Question. A function follows y=3x+5. Which x-value produces output y=-7?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: -7=3x+(5). Subtract the constant 5 on both sides to obtain -12=3x. Divide by 3, yielding x=-4. This reverses the order of the function-machine steps.

Verify from the rule. Put x=-4 into the original function: 3(-4)+(5)=-7. The output matches exactly, so (-4,-7) lies on the line.

Independent transfer. Now change the target output to -4. The required x becomes -3 because one extra x-unit changes y by the fixed slope 3. Explain this without repeating a full equation solve.

39. Compare two linear rules 13: fixed output shift

Question. Compare y=3x+5 with y=3x+7. What changes at input x=-1, and do the graphs have the same gradient?

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 11 and 13, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 4 but unchanged intercept 5. Explain how that changes the rate per input rather than moving every output by a fixed amount.

40. Function-table clinic 14: y=-4x+8

Question. Using y=-4x+8, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by -4, then adds the signed constant 8. At x=-2, y=-4(-2)+(8)=16. At x=0, y=8. At x=3, y=-4. The ordered pairs are (-2,16), (0,8), (3,-4).

The first error to diagnose. A child might put output 16 into the x-column, turning the pair into (16,-2). Another might omit brackets for negative x, writing -4−2 as subtraction rather than -4×(−2).

Verify from the rule. The output change between x=-2 and x=0 is -8 over 2 input units, a rate of -4 per input unit. Between x=0 and x=3, the change is -12 over 3 input units, the same slope -4. A linear function has constant rate of change.

Independent transfer. Hide the table and calculate y at x=4, which should be -8. Explain whether y rises or falls as x grows: the sign of the coefficient -4 answers that question before a calculator is used.

41. Reverse function clinic 14: find the input

Question. A function follows y=-4x+8. Which x-value produces output y=12?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 12=-4x+(8). Subtract the constant 8 on both sides to obtain 4=-4x. Divide by -4, yielding x=-1. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=12 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=-1 into the original function: -4(-1)+(8)=12. The output matches exactly, so (-1,12) lies on the line.

Independent transfer. Now change the target output to 8. The required x becomes 0 because one extra x-unit changes y by the fixed slope -4. Explain this without repeating a full equation solve.

42. Compare two linear rules 14: fixed output shift

Question. Compare y=-4x+8 with y=-4x+10. What changes at input x=-1, and do the graphs have the same gradient?

Did you know? A function turns an input into one output under a specified rule. At x=-1, the original gives 12 and the new rule gives 14. Every output of the second is 2 higher for the same input because only the constant term changed. Both rules have gradient -4; their y-intercepts are (0,8) and (0,10).

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 0 and 2, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient -3 but unchanged intercept 8. Explain how that changes the rate per input rather than moving every output by a fixed amount.

43. Function-table clinic 15: y=4x+6

Question. Using y=4x+6, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 4, then adds the signed constant 6. At x=-2, y=4(-2)+(6)=-2. At x=0, y=6. At x=3, y=18. The ordered pairs are (-2,-2), (0,6), (3,18).

The first error to diagnose. A child might put output -2 into the x-column, turning the pair into (-2,-2). Another might omit brackets for negative x, writing 4−2 as subtraction rather than 4×(−2).

Independent transfer. Hide the table and calculate y at x=4, which should be 22. Explain whether y rises or falls as x grows: the sign of the coefficient 4 answers that question before a calculator is used.

44. Reverse function clinic 15: find the input

Question. A function follows y=4x+6. Which x-value produces output y=14?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 14=4x+(6). Subtract the constant 6 on both sides to obtain 8=4x. Divide by 4, yielding x=2. This reverses the order of the function-machine steps.

Verify from the rule. Put x=2 into the original function: 4(2)+(6)=14. The output matches exactly, so (2,14) lies on the line.

Independent transfer. Now change the target output to 18. The required x becomes 3 because one extra x-unit changes y by the fixed slope 4. Explain this without repeating a full equation solve.

45. Compare two linear rules 15: fixed output shift

Question. Compare y=4x+6 with y=4x+8. What changes at input x=-1, and do the graphs have the same gradient?

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 14 and 16, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 5 but unchanged intercept 6. Explain how that changes the rate per input rather than moving every output by a fixed amount.

46. Function-table clinic 16: y=-5x−1

Question. Using y=-5x−1, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by -5, then adds the signed constant -1. At x=-2, y=-5(-2)+(-1)=9. At x=0, y=-1. At x=3, y=-16. The ordered pairs are (-2,9), (0,-1), (3,-16).

The first error to diagnose. A child might put output 9 into the x-column, turning the pair into (9,-2). Another might omit brackets for negative x, writing -5−2 as subtraction rather than -5×(−2).

Verify from the rule. The output change between x=-2 and x=0 is -10 over 2 input units, a rate of -5 per input unit. Between x=0 and x=3, the change is -15 over 3 input units, the same slope -5. A linear function has constant rate of change.

Independent transfer. Hide the table and calculate y at x=4, which should be -21. Explain whether y rises or falls as x grows: the sign of the coefficient -5 answers that question before a calculator is used.

47. Reverse function clinic 16: find the input

Question. A function follows y=-5x−1. Which x-value produces output y=19?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 19=-5x+(-1). Subtract the constant -1 on both sides to obtain 20=-5x. Divide by -5, yielding x=-4. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=19 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=-4 into the original function: -5(-4)+(-1)=19. The output matches exactly, so (-4,19) lies on the line.

Independent transfer. Now change the target output to 14. The required x becomes -3 because one extra x-unit changes y by the fixed slope -5. Explain this without repeating a full equation solve.

48. Compare two linear rules 16: fixed output shift

Question. Compare y=-5x−1 with y=-5x+1. What changes at input x=-1, and do the graphs have the same gradient?

Did you know? A function turns an input into one output under a specified rule. At x=-1, the original gives 4 and the new rule gives 6. Every output of the second is 2 higher for the same input because only the constant term changed. Both rules have gradient -5; their y-intercepts are (0,-1) and (0,1).

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are -11 and -9, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient -4 but unchanged intercept -1. Explain how that changes the rate per input rather than moving every output by a fixed amount.

49. Function-table clinic 17: y=2x

Question. Using y=2x, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 2, then adds the signed constant 0. At x=-2, y=2(-2)+(0)=-4. At x=0, y=0. At x=3, y=6. The ordered pairs are (-2,-4), (0,0), (3,6).

The first error to diagnose. A child might put output -4 into the x-column, turning the pair into (-4,-2). Another might omit brackets for negative x, writing 2−2 as subtraction rather than 2×(−2).

Independent transfer. Hide the table and calculate y at x=4, which should be 8. Explain whether y rises or falls as x grows: the sign of the coefficient 2 answers that question before a calculator is used.

50. Reverse function clinic 17: find the input

Question. A function follows y=2x. Which x-value produces output y=-2?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: -2=2x+(0). Subtract the constant 0 on both sides to obtain -2=2x. Divide by 2, yielding x=-1. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=-2 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=-1 into the original function: 2(-1)+(0)=-2. The output matches exactly, so (-1,-2) lies on the line.

Independent transfer. Now change the target output to 0. The required x becomes 0 because one extra x-unit changes y by the fixed slope 2. Explain this without repeating a full equation solve.

51. Compare two linear rules 17: fixed output shift

Question. Compare y=2x with y=2x+2. What changes at input x=-1, and do the graphs have the same gradient?

Did you know? A function turns an input into one output under a specified rule. At x=-1, the original gives -2 and the new rule gives 0. Every output of the second is 2 higher for the same input because only the constant term changed. Both rules have gradient 2; their y-intercepts are (0,0) and (0,2).

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 4 and 6, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 3 but unchanged intercept 0. Explain how that changes the rate per input rather than moving every output by a fixed amount.

52. Function-table clinic 18: y=-3x−4

Question. Using y=-3x−4, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by -3, then adds the signed constant -4. At x=-2, y=-3(-2)+(-4)=2. At x=0, y=-4. At x=3, y=-13. The ordered pairs are (-2,2), (0,-4), (3,-13).

The first error to diagnose. A child might put output 2 into the x-column, turning the pair into (2,-2). Another might omit brackets for negative x, writing -3−2 as subtraction rather than -3×(−2).

Independent transfer. Hide the table and calculate y at x=4, which should be -16. Explain whether y rises or falls as x grows: the sign of the coefficient -3 answers that question before a calculator is used.

53. Reverse function clinic 18: find the input

Question. A function follows y=-3x−4. Which x-value produces output y=-10?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: -10=-3x+(-4). Subtract the constant -4 on both sides to obtain -6=-3x. Divide by -3, yielding x=2. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=-10 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=2 into the original function: -3(2)+(-4)=-10. The output matches exactly, so (2,-10) lies on the line.

Independent transfer. Now change the target output to -13. The required x becomes 3 because one extra x-unit changes y by the fixed slope -3. Explain this without repeating a full equation solve.

54. Compare two linear rules 18: fixed output shift

Question. Compare y=-3x−4 with y=-3x−2. What changes at input x=-1, and do the graphs have the same gradient?

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are -10 and -8, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient -2 but unchanged intercept -4. Explain how that changes the rate per input rather than moving every output by a fixed amount.

55. Function-table clinic 19: y=6x−3

Question. Using y=6x−3, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 6, then adds the signed constant -3. At x=-2, y=6(-2)+(-3)=-15. At x=0, y=-3. At x=3, y=15. The ordered pairs are (-2,-15), (0,-3), (3,15).

The first error to diagnose. A child might put output -15 into the x-column, turning the pair into (-15,-2). Another might omit brackets for negative x, writing 6−2 as subtraction rather than 6×(−2).

Verify from the rule. The output change between x=-2 and x=0 is 12 over 2 input units, a rate of 6 per input unit. Between x=0 and x=3, the change is 18 over 3 input units, the same slope 6. A linear function has constant rate of change.

Independent transfer. Hide the table and calculate y at x=4, which should be 21. Explain whether y rises or falls as x grows: the sign of the coefficient 6 answers that question before a calculator is used.

56. Reverse function clinic 19: find the input

Question. A function follows y=6x−3. Which x-value produces output y=-27?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: -27=6x+(-3). Subtract the constant -3 on both sides to obtain -24=6x. Divide by 6, yielding x=-4. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=-27 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=-4 into the original function: 6(-4)+(-3)=-27. The output matches exactly, so (-4,-27) lies on the line.

Independent transfer. Now change the target output to -21. The required x becomes -3 because one extra x-unit changes y by the fixed slope 6. Explain this without repeating a full equation solve.

57. Compare two linear rules 19: fixed output shift

Question. Compare y=6x−3 with y=6x−1. What changes at input x=-1, and do the graphs have the same gradient?

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 9 and 11, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 7 but unchanged intercept -3. Explain how that changes the rate per input rather than moving every output by a fixed amount.

58. Function-table clinic 20: y=-6x+2

Question. Using y=-6x+2, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by -6, then adds the signed constant 2. At x=-2, y=-6(-2)+(2)=14. At x=0, y=2. At x=3, y=-16. The ordered pairs are (-2,14), (0,2), (3,-16).

The first error to diagnose. A child might put output 14 into the x-column, turning the pair into (14,-2). Another might omit brackets for negative x, writing -6−2 as subtraction rather than -6×(−2).

Verify from the rule. The output change between x=-2 and x=0 is -12 over 2 input units, a rate of -6 per input unit. Between x=0 and x=3, the change is -18 over 3 input units, the same slope -6. A linear function has constant rate of change.

Independent transfer. Hide the table and calculate y at x=4, which should be -22. Explain whether y rises or falls as x grows: the sign of the coefficient -6 answers that question before a calculator is used.

59. Reverse function clinic 20: find the input

Question. A function follows y=-6x+2. Which x-value produces output y=8?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 8=-6x+(2). Subtract the constant 2 on both sides to obtain 6=-6x. Divide by -6, yielding x=-1. This reverses the order of the function-machine steps.

Verify from the rule. Put x=-1 into the original function: -6(-1)+(2)=8. The output matches exactly, so (-1,8) lies on the line.

Independent transfer. Now change the target output to 2. The required x becomes 0 because one extra x-unit changes y by the fixed slope -6. Explain this without repeating a full equation solve.

60. Compare two linear rules 20: fixed output shift

Question. Compare y=-6x+2 with y=-6x+4. What changes at input x=-1, and do the graphs have the same gradient?

Did you know? A function turns an input into one output under a specified rule. At x=-1, the original gives 8 and the new rule gives 10. Every output of the second is 2 higher for the same input because only the constant term changed. Both rules have gradient -6; their y-intercepts are (0,2) and (0,4).

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are -10 and -8, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient -5 but unchanged intercept 2. Explain how that changes the rate per input rather than moving every output by a fixed amount.

61. Function-table clinic 21: y=5x+4

Question. Using y=5x+4, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 5, then adds the signed constant 4. At x=-2, y=5(-2)+(4)=-6. At x=0, y=4. At x=3, y=19. The ordered pairs are (-2,-6), (0,4), (3,19).

The first error to diagnose. A child might put output -6 into the x-column, turning the pair into (-6,-2). Another might omit brackets for negative x, writing 5−2 as subtraction rather than 5×(−2).

Independent transfer. Hide the table and calculate y at x=4, which should be 24. Explain whether y rises or falls as x grows: the sign of the coefficient 5 answers that question before a calculator is used.

62. Reverse function clinic 21: find the input

Question. A function follows y=5x+4. Which x-value produces output y=14?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 14=5x+(4). Subtract the constant 4 on both sides to obtain 10=5x. Divide by 5, yielding x=2. This reverses the order of the function-machine steps.

Verify from the rule. Put x=2 into the original function: 5(2)+(4)=14. The output matches exactly, so (2,14) lies on the line.

Independent transfer. Now change the target output to 19. The required x becomes 3 because one extra x-unit changes y by the fixed slope 5. Explain this without repeating a full equation solve.

63. Compare two linear rules 21: fixed output shift

Question. Compare y=5x+4 with y=5x+6. What changes at input x=-1, and do the graphs have the same gradient?

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 14 and 16, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 6 but unchanged intercept 4. Explain how that changes the rate per input rather than moving every output by a fixed amount.

64. Function-table clinic 22: y=-4x−2

Question. Using y=-4x−2, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by -4, then adds the signed constant -2. At x=-2, y=-4(-2)+(-2)=6. At x=0, y=-2. At x=3, y=-14. The ordered pairs are (-2,6), (0,-2), (3,-14).

The first error to diagnose. A child might put output 6 into the x-column, turning the pair into (6,-2). Another might omit brackets for negative x, writing -4−2 as subtraction rather than -4×(−2).

Independent transfer. Hide the table and calculate y at x=4, which should be -18. Explain whether y rises or falls as x grows: the sign of the coefficient -4 answers that question before a calculator is used.

65. Reverse function clinic 22: find the input

Question. A function follows y=-4x−2. Which x-value produces output y=14?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 14=-4x+(-2). Subtract the constant -2 on both sides to obtain 16=-4x. Divide by -4, yielding x=-4. This reverses the order of the function-machine steps.

Verify from the rule. Put x=-4 into the original function: -4(-4)+(-2)=14. The output matches exactly, so (-4,14) lies on the line.

Independent transfer. Now change the target output to 10. The required x becomes -3 because one extra x-unit changes y by the fixed slope -4. Explain this without repeating a full equation solve.

66. Compare two linear rules 22: fixed output shift

Question. Compare y=-4x−2 with y=-4x. What changes at input x=-1, and do the graphs have the same gradient?

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are -10 and -8, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient -3 but unchanged intercept -2. Explain how that changes the rate per input rather than moving every output by a fixed amount.

67. Function-table clinic 23: y=3x−8

Question. Using y=3x−8, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 3, then adds the signed constant -8. At x=-2, y=3(-2)+(-8)=-14. At x=0, y=-8. At x=3, y=1. The ordered pairs are (-2,-14), (0,-8), (3,1).

The first error to diagnose. A child might put output -14 into the x-column, turning the pair into (-14,-2). Another might omit brackets for negative x, writing 3−2 as subtraction rather than 3×(−2).

Independent transfer. Hide the table and calculate y at x=4, which should be 4. Explain whether y rises or falls as x grows: the sign of the coefficient 3 answers that question before a calculator is used.

68. Reverse function clinic 23: find the input

Question. A function follows y=3x−8. Which x-value produces output y=-11?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: -11=3x+(-8). Subtract the constant -8 on both sides to obtain -3=3x. Divide by 3, yielding x=-1. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=-11 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=-1 into the original function: 3(-1)+(-8)=-11. The output matches exactly, so (-1,-11) lies on the line.

Independent transfer. Now change the target output to -8. The required x becomes 0 because one extra x-unit changes y by the fixed slope 3. Explain this without repeating a full equation solve.

69. Compare two linear rules 23: fixed output shift

Question. Compare y=3x−8 with y=3x−6. What changes at input x=-1, and do the graphs have the same gradient?

Did you know? A function turns an input into one output under a specified rule. At x=-1, the original gives -11 and the new rule gives -9. Every output of the second is 2 higher for the same input because only the constant term changed. Both rules have gradient 3; their y-intercepts are (0,-8) and (0,-6).

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are -2 and 0, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 4 but unchanged intercept -8. Explain how that changes the rate per input rather than moving every output by a fixed amount.

70. Function-table clinic 24: y=x

Question. Using y=x, complete an input–output table at x=-2, 0, 3. Explain which column is input and which is output.

Did you know? A function turns an input into one output under a specified rule. The function takes input x, multiplies it by 1, then adds the signed constant 0. At x=-2, y=1(-2)+(0)=-2. At x=0, y=0. At x=3, y=3. The ordered pairs are (-2,-2), (0,0), (3,3).

The first error to diagnose. A child might put output -2 into the x-column, turning the pair into (-2,-2). Another might omit brackets for negative x, writing 1−2 as subtraction rather than 1×(−2).

Independent transfer. Hide the table and calculate y at x=4, which should be 4. Explain whether y rises or falls as x grows: the sign of the coefficient 1 answers that question before a calculator is used.

71. Reverse function clinic 24: find the input

Question. A function follows y=x. Which x-value produces output y=2?

Did you know? A function turns an input into one output under a specified rule. Replace y with the target: 2=1x+(0). Subtract the constant 0 on both sides to obtain 2=1x. Divide by 1, yielding x=2. This reverses the order of the function-machine steps.

The first error to diagnose. Treating x=2 as an answer ignores the transformation. Dividing by the coefficient before removing the constant changes the relationship, particularly when the intercept is negative.

Verify from the rule. Put x=2 into the original function: 1(2)+(0)=2. The output matches exactly, so (2,2) lies on the line.

Independent transfer. Now change the target output to 3. The required x becomes 3 because one extra x-unit changes y by the fixed slope 1. Explain this without repeating a full equation solve.

72. Compare two linear rules 24: fixed output shift

Question. Compare y=x with y=x+2. What changes at input x=-1, and do the graphs have the same gradient?

The first error to diagnose. A pupil may say the slope changes by two because the new rule has ‘two more’ in it. That difference is a vertical shift, not a change in output per input step.

Verify from the rule. At x=2, the outputs are 2 and 4, again separated by 2. The constant gap across different inputs verifies that the lines are parallel when the gradients match.

Independent transfer. Invent a third rule with coefficient 2 but unchanged intercept 0. Explain how that changes the rate per input rather than moving every output by a fixed amount.

73. Context and data-table clinic 1: A one-off membership fee

Question. An imaginary club charges $7 joining plus $3 per attended session. Write total cost C after n sessions and make a table at 0,2,5.

Did you know? A function turns an input into one output under a specified rule. C=7+3n. At n=0 cost is $7; at two it is $13; at five it is $22. The input counts sessions, not dollars.

The first error to diagnose. Writing C=7n+3 reverses fixed and variable roles.

Verify from the rule. The y-intercept is seven, indicating the charge when input is zero.

Independent transfer. Identify what x measures and what y measures before calculating. Then write a changed input or output and state whether the same constant-rate model still applies under the story’s assumptions. The response should include the appropriate quantity and unit, not merely an unlabelled coordinate.

74. Context and data-table clinic 2: A package with no start-up fee

Question. A machine prints six labels per minute from zero. Make output P at t=0,1,4.

Did you know? A function turns an input into one output under a specified rule. P=6t. Outputs are 0,6,24 labels. Constant rate six labels per minute.

The first error to diagnose. Adding a nonzero intercept invents output before the process begins.

Verify from the rule. At t=4, 6×4=24; the graph passes through origin in the ideal model.

75. Context and data-table clinic 3: A decreasing inventory

Question. A box begins with 50 counters, then five are removed per day. Give expression and count after six days.

Did you know? A function turns an input into one output under a specified rule. C=50−5d; at day six there are 20 counters, assuming no new additions.

The first error to diagnose. Using +5 because the story contains a repeated quantity.

Verify from the rule. A negative slope represents depletion, and the model is meaningful only before stock would become negative.

76. Context and data-table clinic 4: Temperature falling

Question. A made-up temperature model is T=12−2h in degrees Celsius, h hours after a stated moment. Find T at h=4.

Did you know? A function turns an input into one output under a specified rule. T=12−8=4°C. The constant 12 is the starting temperature, and −2 is the change per hour.

The first error to diagnose. Assuming negative gradient means all temperature outputs are negative.

Verify from the rule. At h=0, T=12; at h=5, T=2, consistently falling.

77. Context and data-table clinic 5: Compare fictional mobile plans

Question. Plan A costs $10+$2 per unit, Plan B costs $4+$3 per unit. At five units which is cheaper?

Did you know? A function turns an input into one output under a specified rule. A=20 and B=19, so B is $1 cheaper for five units. The rule changes the better option as usage changes.

The first error to diagnose. Choosing A automatically because its per-unit charge is lower ignores the larger fixed fee.

Verify from the rule. At six units, A=22 and B=22; equal totals occur at different usage.

78. Context and data-table clinic 6: Interpret a rule as a graph

Question. A fictional savings model S=4w+10 has w weeks and S dollars. What does (3,22) mean?

Did you know? A function turns an input into one output under a specified rule. At week three, the model gives 4×3+10=$22. The pair records the week input and savings output.

The first error to diagnose. Calling twenty-two weeks and three dollars reverses the labels.

Verify from the rule. The starting amount is $10 at week zero.

79. Context and data-table clinic 7: A nonproportional straight line

Question. Is C=6+3n a direct proportion of n?

Did you know? A function turns an input into one output under a specified rule. No. A direct proportional model has zero intercept; here C=6 when n=0. It is linear with fixed charge six.

The first error to diagnose. Every straight-line graph is not necessarily direct proportional.

Verify from the rule. Check C/n changes when n changes from one to two.

80. Context and data-table clinic 8: A positive coefficient is rate, not starting amount

Question. For D=40t+5, what are the rate and starting amount?

Did you know? A function turns an input into one output under a specified rule. Slope is 40 units per unit of t; at t=0 the starting quantity is five. Units depend on axis definitions.

The first error to diagnose. Calling 40 the intercept and 5 the slope reverses the quantities.

Verify from the rule. Compare D at t=0 and 1: five then forty-five, a change of forty.

81. Context and data-table clinic 9: Two equal y-values do not mean identical lines

Question. Functions y=2x+3 and y=4x+1 both output five at x=1. Are they the same function?

Did you know? A function turns an input into one output under a specified rule. No. One shared point is not equal behaviour for all inputs. At x=2, their outputs are 7 and 9.

The first error to diagnose. Checking just one row and declaring two rules identical overgeneralises.

Verify from the rule. Evaluate two more inputs to reveal their different slopes.

82. Context and data-table clinic 10: An x-table can use fractional inputs

Question. For y=2x+1, find output when x=0.5.

Did you know? A function turns an input into one output under a specified rule. y=2(0.5)+1=2. Fractional x-inputs are allowed when a numerical model’s domain permits them.

The first error to diagnose. Assuming all graph tables use whole-number x-values only.

Verify from the rule. At x=1.5, y=4, continuing the same rate.

83. Context and data-table clinic 11: The vertical-axis intercept is an ordered pair

Question. For y=−3x+8, find its y-intercept.

Did you know? A function turns an input into one output under a specified rule. x=0 makes the x-term zero, leaving y=8, so intercept is (0,8).

The first error to diagnose. Writing (8,0) places the point on the wrong axis.

Verify from the rule. Substitute x=0 back into the rule.

84. Context and data-table clinic 12: A negative intercept is a legitimate starting value

Question. For y=2x−7, what is the y-intercept?

Did you know? A function turns an input into one output under a specified rule. At x=0, y=−7, so the intercept is (0,−7).

The first error to diagnose. Ignoring the negative sign or assuming graph intercepts cannot be negative.

Verify from the rule. The line crosses the vertical axis below zero.

85. Context and data-table clinic 13: An x-intercept means output zero

Question. Find where y=3x−6 crosses the x-axis.

Did you know? A function turns an input into one output under a specified rule. Set y=0: 3x−6=0, so x=2 and crossing is (2,0).

The first error to diagnose. Setting x=0 instead finds the vertical intercept.

Verify from the rule. Check at x=2 output is exactly zero.

86. Context and data-table clinic 14: A horizontal function has a zero slope

Question. What are slope and y-intercept for y=7?

Did you know? A function turns an input into one output under a specified rule. The output stays seven for every input, so slope is zero and y-intercept is (0,7).

The first error to diagnose. Calling it undefined gradient because the line is flat confuses horizontal and vertical lines.

Verify from the rule. At x=−3 and 3, y remains seven.

87. Context and data-table clinic 15: A vertical line is not a function y=mx+b

Question. Can the relation x=4 be expressed as a single-output rule y=f(x) for every y on that line?

Did you know? A function turns an input into one output under a specified rule. No. At x=4, many y-values occur, so it is not a function of x in the usual vertical-line-test sense.

The first error to diagnose. Assuming any line on the plane must have form y=mx+b.

Verify from the rule. Compare y=4 (a horizontal function) with x=4 (a vertical relation).

88. Context and data-table clinic 16: A scaled graph needs correctly read units

Question. An x-axis ticks by 2 and a y-axis by 5. How should the point (4,10) be plotted?

Did you know? A function turns an input into one output under a specified rule. Move two x-intervals right and two y-intervals up, using the stated scales.

The first error to diagnose. Counting four and ten grid squares ignores axis labels.

Verify from the rule. The ordered pair’s values are unchanged by the visual tick spacing.

89. Context and data-table clinic 17: A table can be inconsistent with one claimed linear rule

Question. The data are x=0,1,2 and y=3,5,8. Is y=2x+3 consistent with every row?

Did you know? A function turns an input into one output under a specified rule. At x=2, the rule predicts 7, not 8. The table does not exactly follow the proposed rule.

The first error to diagnose. Accepting a formula because it matches two rows but not the third.

Verify from the rule. Test each supplied input; consistent modelling uses all data.

90. Context and data-table clinic 18: Solve a target without plotting

Question. For y=−2x+9, what input gives y=1?

Did you know? A function turns an input into one output under a specified rule. Solve 1=−2x+9, so −8=−2x and x=4.

The first error to diagnose. Reading y=1 as if x must also be one.

Verify from the rule. Check −2(4)+9=1.

91. Context and data-table clinic 19: A starting quantity may not be physically negative

Question. An ideal water-tank model W=30−4t gives W=−2 at t=8. Is that a meaningful real tank volume?

Did you know? A function turns an input into one output under a specified rule. No. The linear rule may be useful before the tank empties, but a negative physical volume violates the model’s domain.

The first error to diagnose. Extending an equation indefinitely without considering the context.

Verify from the rule. Solve 30−4t=0 to find theoretical empty time t=7.5, where the model should stop.

92. Context and data-table clinic 20: Tables reveal constant differences

Question. A table gives y=6,10,14,18 at x=0,2,4,6. Find gradient.

Did you know? A function turns an input into one output under a specified rule. Output rises four as x rises two, so slope is 4/2=2. Intercept at x=0 is six: y=2x+6.

The first error to diagnose. Using rise four as the slope without dividing by the input change.

Verify from the rule. Test x=6: 2×6+6=18.

93. Context and data-table clinic 21: A fixed intercept can be checked at zero

Question. A rule is y=−4x+5. A classmate says its intercept is −4. Explain the error.

Did you know? A function turns an input into one output under a specified rule. At x=0, y=5, so intercept is (0,5). −4 is gradient, the change in y per +1 x.

The first error to diagnose. Treating the first printed number as the intercept regardless of x.

Verify from the rule. At x=1 output drops by four to one.

94. Context and data-table clinic 22: An increasing output need not mean proportional

Question. A salary-like imaginary function has starting balance $100 and adds $10 each week. Is savings directly proportional to weeks?

Did you know? A function turns an input into one output under a specified rule. No. S=100+10w is linear but not through origin. Its week-zero value is 100.

The first error to diagnose. Using the word ‘increase’ as automatic evidence of direct proportion.

Verify from the rule. Ratios S/w differ across positive weeks; change per week stays ten.

95. Context and data-table clinic 23: Functions should preserve input unit labels

Question. A bakery prints labels at 12 labels per minute, and a table x=minutes,y=labels. What does gradient mean?

Did you know? A function turns an input into one output under a specified rule. It is 12 labels/min, a rate. The intercept zero means no labels printed at t=0 in this simplified setup.

The first error to diagnose. Reporting 12 km/h because a gradient resembles speed ignores the axis labels.

Verify from the rule. Write the compound unit output units/input units.

96. Context and data-table clinic 24: A graph estimate may not equal an exact formula value

Question. An imperfect hand sketch of y=2x+1 seems to show y=6 at x=2. What does the rule say?

Did you know? A function turns an input into one output under a specified rule. Substitution gives five exactly. The hand sketch may be imprecise and must not override the defined formula.

The first error to diagnose. Trusting a thick pencil line more than an exact algebraic relationship.

Verify from the rule. Calculate at x=0,1,2 and redraw on correct scales.

97. Context and data-table clinic 25: Switching the unknown changes the task

Question. If C=3n+5 and C=20, find n.

Did you know? A function turns an input into one output under a specified rule. Subtract five: 3n=15, so n=5. This is a reverse-input equation, not another substitution into known n.

The first error to diagnose. Using n=20 because twenty was the given total.

Verify from the rule. Check 3×5+5=20.

98. Context and data-table clinic 26: Check a proposed function output

Question. A pupil says y=−2x+3 gives y=9 at x=−3. Is that correct?

Did you know? A function turns an input into one output under a specified rule. Yes. −2×(−3)+3=6+3=9. Brackets protect both negatives.

The first error to diagnose. Treating −2×(−3) as negative six is a signed-product error.

Verify from the rule. At x=0, output is three; as x decreases, y increases by two per step.

Five first-step errors and what they reveal

Swapping x and y in the ordered pair

For y=3x+2 at x=1, the output is five, so the pair is (1,5). Writing (5,1) changes which number represents input. A tutor can use two differently coloured table columns and ask the child to read each pair aloud. Only then plot it on the coordinate grid and verify the table entry matches the selected x.

A negative input is not bracketed

In y=−2x+3 at x=−4, the negative multiplier and negative input produce positive eight, then plus three gives eleven. A student who writes −2−4+3 has silently dropped multiplication. Insist on y=−2(−4)+3 and predict the sign before calculating. Changed-number retests reveal whether bracket notation has become reliable.

The output difference is mistaken for gradient

If x moves by three while y rises twelve, gradient is 12/3=4, not twelve. Ask the learner to name the vertical and horizontal changes separately and divide. A new table with irregular x increments can expose the mistake even when the original consecutive-integer table looked easy.

The intercept is confused with the slope

For y=4x−7, gradient is four and y-intercept is −7. Ask the learner to set x=0 and see −7 directly, then compare outputs at x=0 and x=1 to see an increase of four. These two checks separate starting level from change per step. They should remain distinct in every context.

A fixed fee is mistaken for direct proportion

A service costs $5 plus $3 per activity. A student who models its total as 5n+3 has switched fixed and variable costs. Draw a table including zero activities: the correct rule gives $5 at zero, $8 at one and $11 at two. The table communicates the structure more clearly than a generic instruction to memorise C=an+b.

How a three-pupil Mathematics lesson can use function tables

The immutable eduKateSG Clementi small-group Mathematics reference describes 90-minute three-pupil tutorials near Sixth Avenue MRT. In an appropriate G1 tutorial, one pupil may rebuild signed integer substitution, another fill and graph ordered-pair tables, and a third solve reverse-output word problems. The teaching format is illustrative; confirm actual Secondary 2 G1 class openings and timing directly with the centre.

10 minutes: retrieve signed-number substitution and coordinate order.
20 minutes: tutor demonstrates a function machine, table and labelled graph using one rule.
25 minutes: each pupil works a differentiated rule with checks for intercept and slope.
15 minutes: compare two rules with different intercepts or gradients.
10 minutes: solve an unseen reverse-output question independently.
10 minutes: explain the earliest wrong row and select a delayed retest.

Five short practice sessions for parents

Day 1: fill a five-row table with positive x values.
Day 2: use the same rule with negative inputs and explicit brackets.
Day 3: predict whether the function rises or falls and compute change per unit x.
Day 4: find the input from a given output using equation balance.
Day 5: plot a table for one real-world-style rule, with units and the model’s limits clearly stated.

A useful parent question is ‘What does the rule do first, and what does it do next?’ If the student can give the output for a changed input, explain the row’s units and reverse the rule for a target output, they are using the idea rather than copying a table. Keep the amount of practice manageable and aligned with school assignments.

G1 subject level and the graduating syllabus

MOE’s G1 Secondary Two outline includes functions and graphing, with linear y=ax+b rules and gradients. SEAB’s published 2027 SEC G1 Mathematics K110 syllabus includes linear and quadratic functions over the graduating course. That 2027 paper is for its graduating candidates; a pupil currently in Secondary 2 in 2026 should follow their school topic sequence and later the official specifications for the correct graduating year. This article focuses on the more immediate linear foundation.

Frequently asked questions

Is y=ax+b a direct proportion?

Only when b=0, under the ordinary nonzero-input ratio definition. A nonzero constant b creates a line with a y-intercept away from origin.

What does the first coordinate in a pair mean?

The x-value, the input or horizontal position. The second coordinate gives the y-output or vertical position.

How can gradient be checked from a table?

Divide the difference in outputs by the difference in corresponding inputs, using the same order of rows in both differences.

What if the slope is negative?

The output decreases as x increases. A negative coefficient does not require every output to be negative.

Can a linear model stop being realistic?

Yes. Its mathematical rule may continue for all real x, but an actual context may limit time, usage, inventory or physical quantities. State those constraints.

How can tuition track progress with functions?

Use an independent table, a changed input, a reverse-output problem and a labelled graph after a delay. Each tests the same relationship differently.

Official references and editorial notes

Reader note: Tables, prices, production rules and modelled rates here are original instructional examples, not real services, official examination questions or guaranteed school test topics. School subject-level scope, tuition location, current fees, teachers and group openings require direct verification.

Bring one completed function table and one incorrectly plotted ordered pair to eduKate Singapore and ask the tutor how the child will learn to explain input, output and constant rate independently.

Secondary 2 G1 Mathematics Tutor Bukit Timah | Linear Functions and Input–Output Tables

Quick answer: read the function table before the graph

Find the table or graph skill you need

A 60-second diagnostic using one rule

Why a function machine helps

The x-column is not merely the left side of a table

The constant term is the starting output

The same change repeats across equal input steps

Find the input when an output is given

98 original linear function and table clinics

1. Function-table clinic 1: y=x+3

2. Reverse function clinic 1: find the input

3. Compare two linear rules 1: fixed output shift

4. Function-table clinic 2: y=2x+1

5. Reverse function clinic 2: find the input

6. Compare two linear rules 2: fixed output shift

7. Function-table clinic 3: y=3x−2

8. Reverse function clinic 3: find the input

9. Compare two linear rules 3: fixed output shift

10. Function-table clinic 4: y=−x+4

11. Reverse function clinic 4: find the input

12. Compare two linear rules 4: fixed output shift

13. Function-table clinic 5: y=-2x+7

14. Reverse function clinic 5: find the input

15. Compare two linear rules 5: fixed output shift

16. Function-table clinic 6: y=4x−5

17. Reverse function clinic 6: find the input

18. Compare two linear rules 6: fixed output shift

19. Function-table clinic 7: y=-3x+2

20. Reverse function clinic 7: find the input

21. Compare two linear rules 7: fixed output shift

22. Function-table clinic 8: y=5x

23. Reverse function clinic 8: find the input

24. Compare two linear rules 8: fixed output shift

25. Function-table clinic 9: y=2x−7

26. Reverse function clinic 9: find the input

27. Compare two linear rules 9: fixed output shift

28. Function-table clinic 10: y=-2x−3

29. Reverse function clinic 10: find the input

30. Compare two linear rules 10: fixed output shift

31. Function-table clinic 11: y=x−6

32. Reverse function clinic 11: find the input

33. Compare two linear rules 11: fixed output shift

34. Function-table clinic 12: y=−x−2

35. Reverse function clinic 12: find the input

36. Compare two linear rules 12: fixed output shift

37. Function-table clinic 13: y=3x+5

38. Reverse function clinic 13: find the input

39. Compare two linear rules 13: fixed output shift

40. Function-table clinic 14: y=-4x+8

41. Reverse function clinic 14: find the input

42. Compare two linear rules 14: fixed output shift

43. Function-table clinic 15: y=4x+6

44. Reverse function clinic 15: find the input

45. Compare two linear rules 15: fixed output shift

46. Function-table clinic 16: y=-5x−1

47. Reverse function clinic 16: find the input

48. Compare two linear rules 16: fixed output shift

49. Function-table clinic 17: y=2x

50. Reverse function clinic 17: find the input

51. Compare two linear rules 17: fixed output shift

52. Function-table clinic 18: y=-3x−4

53. Reverse function clinic 18: find the input

54. Compare two linear rules 18: fixed output shift

55. Function-table clinic 19: y=6x−3

56. Reverse function clinic 19: find the input

57. Compare two linear rules 19: fixed output shift

58. Function-table clinic 20: y=-6x+2

59. Reverse function clinic 20: find the input

60. Compare two linear rules 20: fixed output shift

61. Function-table clinic 21: y=5x+4

62. Reverse function clinic 21: find the input

63. Compare two linear rules 21: fixed output shift

64. Function-table clinic 22: y=-4x−2

65. Reverse function clinic 22: find the input

66. Compare two linear rules 22: fixed output shift

67. Function-table clinic 23: y=3x−8

68. Reverse function clinic 23: find the input

69. Compare two linear rules 23: fixed output shift

70. Function-table clinic 24: y=x