# Section
3.19 Transformations of Graphs: An Application of Inverse Functions

In high school you learned about “shifting” graphs, “squeezing” graphs, and
“stretching” graphs. For example, if you focus on the graph of
$y = f(x)$ in Figure 3.19.1, the graph
$y = f(x ‑ 2)$ appears to be identical to it but shifted
to the right.

Figure 3.19.1. The graph in Cartesian coordinates of $
y = f(x ‑ 2)$ appears as the graph of
$y = f(x)$, but shifted two units to the right.

The reason why the graph of $y = f(x ‑ 2)$ appears as
the graph of $y = f(x)$, but shifted, is explained by a general
relationship between the graph of $y = h(x)$ and the graph of
$y = h(k(x))$ for functions *h* and *k*.

Let *h* be any function, and let *k* be any function that has an
inverse function $k^{-1}$. Let $j(x) = h(k(x))$. Envision this
process for plotting a point on the graph of $y = j(x)$, or
$y = h(k(x))$, in Cartesian coordinates.

1. Remind yourself that the point plotted at $x = c$ will have a *
y*-coordinate of $j(c)$, or $h(k(c))$.

2. Start at $x = c$ on the *x*-axis. Move to
$x = k(c)$ on the *x*-axis. This is the value of *x*
on the horizontal axis where we will find the point having the *y*-value
$h(k(c))$.

3. Evaluate *h* at $x = k(c)$ to find the point $(k(c),
h(k(c)))$ on *h*’s graph.

4. Evaluate $k^{-1}(k(c))$ to move back to $x = c$. Plot the point
$(c, h(k(c))$. This will be the point on the graph of
$y = j(x)$ at $x = c$.

5. Think of $k(c)$ as *u* in the domain of *h*’s independent
variable. The point $(u, h(u))$ appears to be moved horizontally to a point
on the graph of *j* by the value of $k^{-1}(u)$.

6. Apply steps 1-5 to every value in the domain of *j*’s independent
variable. The result will seem as if every point $(u, h(u))$ on *h*’s
graph shifts by $k^{-1}(u)$ to be on the graph of *j*.

Follow steps 1-6 as you watch Figure 3.19.2.

*Figure 3.19.2. A visual explanation of
why the graph of $y = h(k(x))$ appears as the graph of
$y = h(u)$, but shifted by $k^{-1}(u)$.*

## Exercise Set 3.19

*(Not yet done: include effects on graphs of all combinations of
operations on functions)*

1. Explain why GC’s displayed graph of $y = \cos(3x - 2)$ is compressed
and shifted the way it is when you compare it to the graph of $y =
\cos(x)$.

2. Let $b(x) = x^2$. Explain why GC’s displayed graph of $
y = b(2x + 1)$ is compressed and shifted the way it is
when you compare it to the graph of $y = b(x)$.

3. There is a function *c*, but we do not know its definition. Let *g*
be defined as $g(u) = u \sin(u)$. Explain in principle how the
graph of $y = c(x)$ is transformed to become the graph of $
y = c(g(x))$.