Review question

How do these transformations change the graph of $f(x)$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9673

Solution

The diagram shows the graph for $y = f(x)$. The curve passes through the origin, and has a maximum point at $(1, 1)$. Sketch, on separate diagrams, the graphs of

1. $y = f(x) + 2$,

… giving the coordinates of the maximum point in each case.

The graph of the function $y = f(x) + 2$ can be obtained from the graph of the original function $f$ by raising the $y$-coordinate of each point on the graph by $2$. In particular, the maximum becomes $(1, 3)$. The new graph looks like this.

1. $y = f(x + 2)$,
Now the graph of the function $y = f(x + 2)$ can be obtained from the graph of the original function $f$ by translating each point on the graph $2$ units to the left.

You might expect a shift to the right, but if you think that the $y$-coordinate associated with $x = 0$ on the old graph is now associated with $x = -2$ on the new graph, you can see it is a translation to the left.

In particular, the maximum becomes $(-1, 1)$. The new graph looks like this.

1. $y = f(2x)$.

The graph of the function $y = f(2x)$ can be obtained from the graph of the original function $f$ by compressing the graph by a scale factor of $2$ in the $x$-direction.

We could describe this as a stretch in the $x$-direction with scale factor $\dfrac{1}{2}$.

In particular, the maximum becomes $(0.5, 1)$. The new graph looks like this.

Can you think of an $f(x)$ that would give something like the starting graph above? You might use graphing software to try a few functions.