Functions and Graphs
What this section tests
Function questions ask you to read and use relationships, not just compute. You will see function notation such as f(x) = 3x − 4, evaluation at a given input, domain and range, slope and intercepts, equations of lines, and graphs of real situations where you must interpret what a slope or an intercept means in context. You may also be asked to recognise whether a graph represents a function at all.
Core rules
- f(2) means substitute 2 for every x. f(a + 1) means substitute the whole expression, using parentheses.
- Slope between two points is m = (y₂ − y₁) ÷ (x₂ − x₁), rise over run, with the points subtracted in the same order both times.
- Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept, the value of y when x = 0.
- Parallel lines share a slope; perpendicular lines have slopes that are negative reciprocals, so m₁ · m₂ = −1.
- On a graph, the domain is the set of x-values covered and the range is the set of y-values. A vertical line may cross a function's graph only once.
- A positive slope means the quantity increases as the input increases; in a real graph, slope is a rate - dollars per hour, feet per second.
Worked example
A line passes through (2, −1) and (6, 7). Find its equation. m = (7 − (−1)) ÷ (6 − 2) = 8 ÷ 4 = 2. Then y = 2x + b. Substitute (2, −1): −1 = 4 + b, so b = −5. The equation is y = 2x − 5. Check with the other point: 2(6) − 5 = 7. Correct.
Traps that cost marks
- Computing run over rise instead of rise over run.
- Subtracting the coordinates in opposite orders, which flips the sign of the slope.
- Confusing the x-intercept with the y-intercept when reading a graph.
- Treating f(x + 2) as a shift up; adding inside the parentheses shifts the graph left.
- Applying outside knowledge to a graph instead of reading the labelled axes.