Chapter 13: Review

The chapter problem, solved

The signal crosses its middle value while rising, then reaches a maximum and minimum before repeating. This matches $y=\sin x$ over one cycle, where $0^\circ\leq x\leq360^\circ$. The maximum occurs at $90^\circ$, the minimum at $270^\circ$, and the pattern repeats every $360^\circ$. A vertical shift changes its middle value; a vertical scale changes its height.

Mixed practice: choose a method and explain why

For each item, name the graph feature or method you will use before answering.

1. Sketch the key features of $y=x^3-4x$. Find its intercepts.
2. State the centre and radius of $(x-3)^2+(y+1)^2=16$.
3. Solve $\sin x=0$ for $0^\circ\leq x\leq360^\circ$.
4. Translate $y=x^2$ two units left and three units up.
5. A radius has gradient $4$. Find the tangent gradient and give a reason.
6. State the asymptotes of $y=1/x$ and $y=\tan x$ from $0^\circ$ to $360^\circ$.
7. The point $(2,5)$ lies on $y=f(x)$. Give its image under $y=f(x)-4$.
8. Which family models repeated equal percentage growth? Give a reason.

What you can now do

• recognise and sketch several non-linear graph families;
• read solutions and features from graphs;
• describe transformations;
• use circle equations and tangent gradients.

Your relevance reflection

Return to your example from the opener. Which graph family fits it best now? Add one reason from the mathematics in this chapter.