Q. 69 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 69–72, the graph of a function $f$ is illustrated. Use the graph of $f$ as the first step toward graphing each of the following functions:

$$\begin{gathered} \left(a\right) F\left(x\right)=f\left(x\right)+3 \\ \left(b\right) G\left(x\right)=f(x+2) \\ \left(c\right) P\left(x\right)=-f\left(x\right) \\ \left(d\right) H\left(x\right)=f(x+1)-2 \\ \left(e\right) Q\left(x\right)=\frac{1}{2}f\left(x\right) \\ \left(f\right) g\left(x\right)=f(-x) \\ \left(g\right) h\left(x\right)=f\left(2x\right) \end{gathered}$$

Step-by-Step Solution

Answer

By using the graph of the function $f$, the graph of $F\left(x\right)=f\left(x\right)+3$ is:

The graph of the function $G\left(x\right)=f(x+2)$ is:

The graph of the function $P\left(x\right)=-f\left(x\right)$ is:

The graph of the function $H\left(x\right)=f(x+1)-2$

The graph of the function $Q\left(x\right)=\frac{1}{2}f\left(x\right)$ is:

The graph of the function $g\left(x\right)=f(-x)$ is:

The graph of the function $h\left(x\right)=f\left(2x\right)$ is:

Part (a) Step 1. Given

The graph:

To graph the function $F\left(x\right)=f\left(x\right)+3$

Part (a) Step 2. Graph the function

The graph of $F\left(x\right)$ is obtained by adding $3$ units to $f\left(x\right)$, so that the graph is shifted vertically upward to $3$ units.

Part (b) Step 1. Given

The graph:

To graph the given function

Part (b) Step 2. Graph the function

Replace $x$ by $x+2$, so that the graph shifted horizontally right to $2$ unit.

Part (c) Step 1. Given

The graph:

To graph the given function

Part (c) Step 2. Graph the function

The graph of the function $P\left(x\right)=-f\left(x\right)$ is reflected about $x-$axis, so, the graph is:

Part (d) Step 1. Given

The graph:

To graph the given function.

Part (d) Step 2. Graph the function

The graph of $H\left(x\right)$ is shifted horizontally right to $1$ unit since $x$ is replaced by $(x+1)$.

And the graph also shifted downward to $2$ units, since $2$ is subtracted from the function.

Part (e) Step 1. Given

The graph:

To graph the given function.

Part (e) Step 2. Graph the function

Since the graph  is multiplied by the factor $\frac{1}{2}$, the graph stretched by the factor $\frac{1}{2}$.

Part (f) Step 1. Given

The graph:

To graph the given function.

Part (f) Step 2. Graph the function

The graph of the function is the reflection of the given function about $y-$axis. So,

Part (g) Step 1. Given

The graph:

To graph the given function.

Part (g) Step 2. Graph the function

The function $h\left(x\right)$ is obtained by each $x-$coordinate multiplied by the factor of $2$.