Chapter T, Problem 4CDT

How are graphs of the functions obtained from the graph of f?

1. (a) y = −f(x)
2. (b) y = 2f(x) − 1
3. (c) y = f(x − 3) + 2

To determine

To find: The process to find the function $y=-f\left(x\right)$ from the function $f\left(x\right)$.

Explanation of Solution

According to the reflection of the function, the graph of the function $y=-f\left(x\right)$ is obtained if the graph of $y=f\left(x\right)$ is reflected vertically about x-axis.

Thus, the function $y=f\left(x\right)$ has to be reflected in order to obtain the function $y=-f\left(x\right)$.

To determine

To find: The process to find the function $y=2f\left(x\right)-1$ from the function $f\left(x\right)$.

Explanation of Solution

According to the magnification of the function the graph of the function $y=kf\left(x\right)$ is obtained if the graph of $y=f\left(x\right)$ is shrunken by k times.

Thus, the function $y=2f\left(x\right)$ is obtained if the graph of $y=f\left(x\right)$ is shrunken by k times.

According to the translation and reflection of the function the graph of the function $y=f\left(x\right)+k$ is obtained if the graph of $y=f\left(x\right)$ is translated upward by k units.

The graph of the function $y=2f\left(x\right)-1$ is obtained if the graph of $y=2f\left(x\right)$ is translated downward by 1 units.

To determine

To find: The process to find the function $y=f\left(x-3\right)+2$ from the function $f\left(x\right)$.

Explanation of Solution

The graph of the function $y=f\left(x-h\right)$ is obtained if the graph of $y=f\left(x\right)$ is translated right by h units.

Thus, the graph of the function $y=f\left(x-3\right)$ is obtained if the graph of $y=f\left(x\right)$ is translated right by 3 units.

The graph of the function $y=f\left(x\right)+k$ is obtained if the graph of $y=f\left(x\right)$ is translated upward by k units.

Thus, the graph of the function $y=f\left(x-3\right)+2$ is obtained if the graph of $y=f\left(x-3\right)$ is translated upward by 2 units.