Functions, Graphs and Modelling
Core concepts
A function assigns exactly one output to each input; the vertical line test confirms this on a graph. Function notation f(x) means the output when the input is x, so f(3) requires substituting 3 everywhere x appears. Understand domain (allowed inputs) and range (possible outputs), and recognise common families: linear (constant rate), quadratic (parabola, symmetric, vertex), exponential (repeated multiplication, rapid growth or decay), and absolute value (V-shape). Transformations follow patterns: f(x) + k shifts up k units, f(x − h) shifts right h units, and a·f(x) stretches vertically when |a| > 1. Modelling means translating a real situation into a function: identify the starting value (intercept) and the rate (slope or growth factor), then interpret the output in context.
Traps to expect
- Confusing f(x + 2) (shift left) with f(x) + 2 (shift up).
- Treating a relation as a function when one input has two outputs.
- Mixing up the domain and range, especially for quadratics where range depends on the vertex.
- For exponential models, using the percent as the growth factor instead of 1 + r (e.g. 5% growth is ×1.05, not ×0.05).
- Reading a graph's intercept as the rate rather than the starting value.
Worked example
A phone plan costs $20 plus $0.10 per minute. Model cost as C(m) = 20 + 0.10m. For 150 minutes: C(150) = 20 + 15 = $35. Here 20 is the fixed start (y-intercept) and 0.10 is the per-minute rate (slope).
Rule of thumb
When modelling, name the input and output in words first, then attach the numbers; this prevents swapping rate and starting value.