Chapter 2.5, Problem 65AYU

In Problems 63-66, the graph of a function f is illustrated. Use the graph of f as the first step toward graphing each of the following functions:

a. $F\left(x\right)\text{ = }f\left(x\right)\text{ + 3}$

b. $G\left(x\right)\text{ = }f\left(x\text{ + 2}\right)$

c. $P\left(x\right)\text{ = -}f\left(x\right)$

d. $H\left(x\right)\text{ = }f\left(x\text{ + 1}\right)\text{ - 2}$

e. $Q\left(x\right)\text{ = }\frac{1}{2}f\left(x\right)$

f. $g\left(x\right)\text{ = -}f\left(x\right)$

g. $h\left(x\right)\text{ = }f\left(2x\right)$

65.

To determine

To graph: The each of the following functions:

a. $F\left(x\right)=f\left(x\right)+3$

For the graph function $f\left(x\right)$,

Explanation of Solution

Graph:

a. $F\left(x\right)=f\left(x\right)+\text{ 3}$

To obtain the graph of $F\left(x\right)=f\left(x\right)+\text{ 3}$ add 3 from each $y\text{-coordinate}$ on the graph of $f\left(x\right)$, that it is shifted up 3 units.

$f\left(x\right)+3$     vertically shift up 3 units

To determine

To graph: The each of the following functions:

b. $G\left(x\right)=f\left(x+2\right)$

For the graph function $f\left(x\right)$,

Explanation of Solution

Graph:

b. $G\left(x\right)=f\left(x+2\right)$

To obtain the graph of $f\left(x\right)=G\left(x\right)=f\left(x+2\right)$, replace $x$ by $x+2$ from each $y\text{-coordinate}$ on the graph of $f\left(x\right)$, that it is shifted left 2 units.

$G\left(x\right)=f\left(x+2\right)$     horizontal shift left 2 units

To determine

To graph: The each of the following functions:

c. $P\left(x\right)=-f\left(x\right)$

For the graph function $f\left(x\right)$,

Explanation of Solution

Graph:

c. $P\left(x\right)=-f\left(x\right)$

To obtain the graph of $P\left(x\right)=-f\left(x\right)$, multiply by $-1$ on the graph of $f\left(x\right)$, that it is reflection about the $x\text{-axis}$.

$P\left(x\right)=-f\left(x\right)$     multiply by $-1$, reflection about the $x\text{-axis}$

To determine

To graph: The each of the following functions:

d. $H\left(x\right)=f\left(x+1\right)-2$

For the graph function $f\left(x\right)$,

Explanation of Solution

Graph:

d. $H\left(x\right)=f\left(x+1\right)-2$

Now use the following steps to obtain the graph of $H(x)$.

Step 1: To obtain the graph of $f\left(x+1\right)$, replace $x$ by $x+1$ from each $y\text{-coordinate}$ on the graph of $f\left(x\right)$, that it is shifted left 1 unit.

$f\left(x\right)=-{\left(x+1\right)}^3$     replace $x$ by $x+1$; horizontal shift left 1 unit

Step 2: To obtain the graph of $H\left(x\right)=f\left(x+1\right)-2$, subtract 2 from each $y\text{-coordinate}$ on the graph of $f\left(x+1\right)$, that it is shifted down 2 units.

$H\left(x\right)=f\left(x+1\right)-2$     subtract 2, vertically shift down 2 unit

To determine

To graph: The each of the following functions:

e. $Q\left(x\right)= \frac{1}{2}f\left(x\right)$

For the graph function $f\left(x\right)$,

Explanation of Solution

Graph:

e. $Q\left(x\right)= \frac{1}{2}f\left(x\right)$

To obtain the graph of $Q\left(x\right)= \frac{1}{2}f\left(x\right)$, multiply each $y\text{-coordinate}$ of the graph of $f\left(x\right)$ by $\frac{1}{2}$ that it is vertically stretched by the factor of $\frac{1}{2}$.

$Q\left(x\right)= \frac{1}{2}f\left(x\right)$     multiply by $\frac{1}{2}$; vertical stretch by a factor of $\frac{1}{2}$

To determine

To graph: The each of the following functions:

f. $g\left(x\right)=f\left(-x\right)$

For the graph function $f\left(x\right)$,

Explanation of Solution

Graph:

f. $g\left(x\right)=f\left(-x\right)$

To obtain the graph of $g\left(x\right)=f\left(-x\right)$, replace $x$ by $-x$ of the graph of $f\left(x\right)$, that it is reflection about the $y\text{-axis}$.

$g\left(x\right)=f\left(-x\right)$     replace $x$ by $-x$, reflect about the $y\text{-axis}$

To determine

To graph: The each of the following functions:

g. $h\left(x\right)=f\left(2x\right)$

For the graph function $f\left(x\right)$,

Explanation of Solution

Graph:

g. $h\left(x\right)=f\left(2x\right)$

To obtain the graph of $h\left(x\right)=f\left(2x\right)$, replace $x$ by $2x$ of the graph of $f\left(x\right)$; that it is horizontally stretched by the factor of $\frac{1}{2}$.

$h\left(x\right)=f\left(2x\right)$     replace $x$ by $2x$ horizontally stretched by a factor of $\frac{1}{2}$