Function transformations are easier when every change is classified as acting on the input or on the output. Outside the function, you change output values. Inside the function, you change which input produces a familiar output.
This guide develops translations, reflections, stretches and compressions from that principle, then combines them without relying on a collection of disconnected arrow rules.
1. Begin with y=f(x)
Think of a point (a,b) on y=f(x). This means f(a)=b. A transformation rule can be understood by asking where that known point must move.
2. Vertical translation: y=f(x)+k
Adding k after the function adds k to every output. The point (a,b) becomes (a,b+k). Positive k moves the graph upward; negative k moves it downward.
Example: y=x² has turning point (0,0). Therefore y=x²+5 has turning point (0,5).
3. Horizontal translation: y=f(x−h)
To reproduce the old output b=f(a), the new input must satisfy x−h=a, so x=a+h. Thus the point moves from (a,b) to (a+h,b). Positive h moves the graph right.
This explains the apparently reversed sign: y=f(x−3) moves right by 3 because x must be 3 larger to feed the old input into f.
4. Reflection in the horizontal axis: y=−f(x)
Every output changes sign. A point (a,b) becomes (a,−b). Horizontal-axis crossings remain in the same positions because zero changes to zero.
5. Reflection in the vertical axis: y=f(−x)
To reproduce f(a), the new input must satisfy −x=a, so x=−a. The point (a,b) becomes (−a,b). This reflects the graph in the vertical axis.
6. Vertical scaling: y=af(x)
Every output is multiplied by a. A point (x,y) becomes (x,ay). When |a|>1, vertical distances from the horizontal axis increase. When 0<|a|<1, they decrease. A negative a also reflects the graph in the horizontal axis.
7. Horizontal scaling: y=f(bx)
If f(a)=c, then f(bx)=c when bx=a, so x=a/b. Horizontal coordinates are divided by b. For b>1, familiar features occur closer to the vertical axis.
This is why y=sin(2x) has half the period of y=sin x. The same input cycle is completed in half the x-distance.
8. Worked example from a parabola
Start with y=x². Transform it to y=−2(x−3)²+4.
- x−3 moves the graph right by 3.
- The factor 2 doubles vertical distances from the horizontal axis.
- The negative sign reflects outputs.
- +4 moves the result upward by 4.
The turning point is (3,4), and the parabola opens downward. Substituting x=3 gives y=4, confirming the turning point immediately.
9. Order matters when transformations combine
Compare 2f(x)+3 with 2[f(x)+3]. The first doubles the original output and then adds 3. The second adds 3 first and then doubles the entire result, giving 2f(x)+6.
Brackets reveal what the multiplier acts on. Do not describe a transformation sequence until the algebraic grouping is clear.
10. Domain and range can change
If f has domain x≥0, then y=f(x−4) requires x−4≥0, so its domain is x≥4. If f has range y≥2, then y=3f(x)−1 has range y≥5 when the multiplier 3 is positive.
For a negative multiplier, inequalities reverse when determining a transformed range. Track actual output values rather than relying on a visual slogan.
11. Use invariant features as checks
A vertical translation does not change horizontal distances between corresponding features. A horizontal translation does not change output differences. Reflection in the horizontal axis preserves horizontal-axis crossings. These invariants can expose a transformation error quickly.
12. Practice
- Describe the transformation from y=f(x) to y=f(x)+6.
- Describe y=f(x−4).
- A point (2,5) lies on y=f(x). Where does it move on y=−f(x)?
- Where does (2,5) move on y=f(−x)?
- If y=sin x has period 2π, state the period of y=sin(3x).
- State the turning point of y=3(x+2)²−7.
13. Answers
- Translate upward by 6.
- Translate right by 4.
- (2,−5).
- (−2,5).
- 2π/3.
- (−2,−7).
Continue with Trigonometric Functions and Graphs, the Functions guide, or return to the Additional Mathematics Hub.