13.3 Transformations of graphs — Learn

Prerequisites: You can plot points and substitute into a function rule. You can describe movement on coordinate axes.

Objectives: You will write rules for translations, reflections and stretches. You will track points and describe how a graph changes.

A transformation changes a graph's position, direction or shape. Write the original rule as $y=f(x)$.

Translations

In $y=f(x)+a$, each output rises by $a$. In $y=f(x-a)$, the graph moves right by $a$. The sign inside the input appears opposite to the direction.

Example 1. Translate $y=x^2$ three units right and two units down.

Replace $x$ by $x-3$ to move right:

Subtract two from each output to move down:

The turning point moves from $(0,0)$ to $(3,-2)$.

Reflections

Reflecting in the $x$-axis changes each output's sign, giving $y=-f(x)$. Reflecting in the $y$-axis changes each input's sign, giving $y=f(-x)$.

Example 2. The point $(2,5)$ lies on $y=f(x)$. Find its image on $y=f(x)+3$.

The input stays at $2$:

The output increases by $3$:

So the image is $(2,8)$.

Watch out. In $f(x-3)$, the graph moves right by $3$. Check with a known point: the original point at $x=0$ appears at $x=3$.

Progress check

The chapter problem's signal is moved up by $4$. Predict how every point changes, then state the new rule if it was $y=f(x)$.

You should now be able to

• write rules for translations, reflections and stretches;
• describe how a point changes under a transformation;
• explain the direction of horizontal translations;
• reflect a graph in either coordinate axis.