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 $k^{-1}$. Envision this process for plotting a point on the graph of $y = h(k(x))$, in Cartesian coordinates.

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

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 $h(k(c))$.

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

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 $y = h(k(x))$ at $x = c$.

Think of $k(c)$ as u in the domain of h. The point $(u, h(u))$ appears to be moved horizontally to a point on the graph of $y=h(k(x))$ by the value of $k^{-1}(u)$.

Apply steps 1-5 to every value in the domain of $h(u)$, where $u=k(x)$. The result will seem as if every point $\left(u, h(u)\right)$ on the graph of $y=h(u)$, where $u=k(x)$, shifts by $k^{-1}(u)$ to be on the graph of $y=h\left(k(x)\right)$.

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)

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 $y = \cos(x)$.

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 $y = b(x)$.

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))$.