What you need to know
Function questions test both algebra and interpretation. You should be able to evaluate a function, follow transformations and connect algebraic equations with points or intersections on a graph.
- Use function notation accurately, including inputs and outputs.
- Recognise translations, reflections and stretches from transformed equations.
- Interpret intersections as simultaneous solutions.
- Check domain restrictions when finding inverses or solving transformed equations.
Key mathematics
Translation
y=f(x-a)+b
Shift right by a and up by b.Vertical scale
y=kf(x)
Multiply every y-coordinate by k.Horizontal scale
y=f(kx)
Horizontal scale factor is 1/k.Worked example
If $y=f(x)$ is shifted 3 units right and 2 units down, the transformed graph is $$y=f(x-3)-2.$$ A point $(p,q)$ on the original graph becomes $(p+3,q-2)$.
EXAM TECHNIQUE
Exam tips
- Describe transformations in a fixed order: horizontal effect, vertical effect, then any reflection/stretch.
- For $f(kx)$ remember the horizontal effect is reciprocal.
- Use a sketch to check whether a transformation direction makes sense.
WATCH OUT
Common mistakes
- Saying $f(x-3)$ moves the graph left instead of right.
- Treating $f(2x)$ as a horizontal stretch by factor 2.
- Confusing $f^{-1}(x)$ with $1/f(x)$.
Frequently asked questions
What does f of x mean?
It names the output of a function when the input is x.
How do I remember horizontal transformations?
Changes inside the function act in the opposite horizontal direction, while changes outside act directly vertically.
Is an inverse function the reciprocal?
No. The inverse function reverses the mapping; it is not the same as one divided by f of x.