Building blocks

## Problem

In Mapping a function, we explored the mapping diagrams of linear functions such as $f(x)=3x$ and $f(x)=2x+1$. Here, we’ll do the same for a familiar non-linear function, considering how it compares to linear functions.

What does the mapping diagram of the function $f(x)=x^2$ look like?

• What does the mapping diagram look like if it is centred on the arrow from $0$ to $f(0)$, at a scale of $1$ unit per tick-mark?

• What if it is centred at some other arrow, say from $1$ to $f(1)$ or $-1$ to $f(-1)$ or $\frac{3}{2}$ to $f\bigl(\frac{3}{2}\bigl)$?

• What does the mapping diagram look like if it is centred on the arrow from $1$ to $f(1)$, but this time zoomed in to $0.1$ or $0.01$ units per tick-mark?

• What if it is instead centred on some other arrow and then zoomed in?

• Can you describe what you observe?

• In what ways are the mapping diagrams of the function $f(x)=x^2$ similar to those of a linear function, and in what ways are they different?

You can download large or small blank mapping diagrams for this, or you can use the GeoGebra applet below (which is the same as the first applet in Mapping a function).

How does this relate to the graph of the function $y=x^2$? You might want to think about the resource Zooming in as you consider this question.

To enter the function $x^2$ into this applet, type x^2.