A function can be written as an equation, drawn as a graph, listed in a table or described in words. These are not four separate mathematical objects. They are different representations of the same input-output relationship.
Strong Additional Mathematics students learn to move between representations because one form often reveals information that another hides. The skill is not merely drawing a graph from an equation; it is recognising what remains invariant when the representation changes.
1. Start with one simple function
Take f(x)=2x+1. The equation says multiply the input by 2 and add 1. A table may show (0,1), (1,3), (2,5). The graph is a straight line through those points. A verbal description might say: “the output starts at 1 when the input is zero and increases by 2 for every increase of 1 in the input.”
All four descriptions encode the same gradient 2 and intercept 1.
2. What an equation reveals quickly
An equation is compact and supports exact algebra. In y=3(x−2)²−4, the completed-square form reveals a turning point at (2,−4), an axis x=2 and a positive vertical scale factor.
The expanded form y=3x²−12x+8 hides the turning point but makes the polynomial coefficients explicit. Equivalent equations can therefore emphasise different features of the same graph.
3. What a graph reveals quickly
A graph makes shape, intersections, turning behaviour and approximate ranges visible. It can show whether two equations have common solutions by displaying intersections.
But a graph drawn from limited points can mislead. A smooth curve must be justified by the function structure, not by joining a few values in whichever way looks plausible.
4. What a table reveals—and what it can hide
A table provides exact or measured input-output pairs. Differences in a linear table may reveal a constant rate of change. A quadratic table with equally spaced inputs has constant second differences.
However, a finite table does not uniquely determine every possible function without additional assumptions. Many different functions can pass through the same finite collection of points.
5. What a verbal description contributes
Words provide context and relationships: “the quantity decreases at a constant rate,” “the maximum occurs when x=4,” or “doubling the input multiplies the output by four.” These statements suggest structures that can be represented algebraically.
The danger is translating a word too quickly. “Increases by 5” and “increases by 5%” describe additive and multiplicative changes respectively.
6. Worked connection: quadratic equation to graph and table
Consider y=(x−1)²−4. The equation gives turning point (1,−4). Setting y=0 gives (x−1)²=4, so x=−1 or 3. The graph therefore crosses the horizontal axis at (−1,0) and (3,0).
A table around the turning point gives y-values 0, −3, −4, −3, 0 for x=−1,0,1,2,3. The symmetry in the table matches the graph’s axis x=1.
7. Worked connection: description to equation
Suppose a straight-line function has output 7 when x=0 and increases by 3 whenever x increases by 1. The intercept is 7 and gradient is 3, so y=3x+7.
At x=4 the equation predicts y=19. A table or graph should agree. If it does not, one representation has been constructed incorrectly.
8. Worked connection: graph features to a cosine model
Suppose a sinusoidal graph has maximum 6, minimum 2 and period π, with a maximum at x=0. Its centre is 4 and amplitude is 2. A cosine model with period π requires an inside multiplier 2. One suitable equation is y=2cos(2x)+4.
The graph features determine the parameters. Substitution at x=0 gives 6, providing an immediate check.
9. Representation switching can expose errors
If algebra gives a quadratic with two real roots but your sketch never crosses the horizontal axis, the two representations disagree. If a table for a claimed increasing linear function contains decreasing outputs, investigate the data or equation.
Agreement across representations is a checking tool. It is especially useful when one calculation is long enough that repeating it would be inefficient.
10. Choose the representation that serves the target
- Use factorised form to see roots.
- Use completed-square form to see a quadratic turning point.
- Use a graph to see intersections and global shape.
- Use a table to inspect selected values or transformed data.
- Use words to state meaning, assumptions and context.
11. Practice
- For y=4x−3, state the gradient, intercept and two table values.
- For y=(x+2)²−5, state the turning point and axis of symmetry.
- A line passes through (0,−2) and increases by 5 per unit x. Write its equation.
- A sinusoidal graph has maximum 9 and minimum 3. Find its centre and amplitude.
- Explain why five table points alone do not prove that the underlying function is linear unless additional structure is known.
12. Answers
- Gradient 4; intercept −3; for example (0,−3) and (1,1).
- Turning point (−2,−5); axis x=−2.
- y=5x−2.
- Centre 6; amplitude 3.
- Other non-linear functions can pass through a finite set of points; the linear model requires a stated or justified assumption about the relationship between them.
13. What mastery looks like
A student has strong representation control when they can extract the same mathematical features from equations, graphs, tables and descriptions; choose the most useful form for a target; and use a second representation to check the first.
Continue with How to Read an A-Math Function Before Solving It, Function Transformations, or return to the Additional Mathematics Hub.