Chapter 1.4, Problem 22E

a. $$

To determine

To graph:

To sketch the graph of $y=f\left(x\right)-1$ .

Answer to Problem 22E

The graph is given by

  

Explanation of Solution

Given information:

The given function is $y=f\left(x\right)-1$ and the given $f$ graph is

  

Graph:

  

Here, we transform the graph $f\left(x\right)$ by $1$ units downwards.

Calculation of Graph:

Consider $y=f\left(x\right)-1$

Values of x	Values of y
$-2$	$3$
$0$	$2$
$1$	$-1$
$3$	$-2$

Using the above table, the graph can be plotted.

Interpretation:

In the graph above, the black curve is the graph of $y=f\left(x\right)$ and the red curve is the graph of $y=f\left(x\right)-1$ .

Conclusion:

Therefore, we have sketched the graph of $y=f\left(x\right)-1$ .

b. $$

To determine

To graph:

To sketch the graph of $y=f\left(x\right)+3$ .

Answer to Problem 22E

The graph of $y=f\left(x\right)+3$ is given by

  

Explanation of Solution

Given information:

The given function is $y=f\left(x\right)+3$ and the given $f$ graph is

  

Graph:

  

Here, we transform the graph $f\left(x\right)$ by $1$ units downwards.

Calculation of Graph:

Consider $y=f\left(x\right)+3$

Values of x	Values of y
$-2$	$7$
$0$	$6$
$1$	$3$
$3$	$2$

Using the above table, the graph can be plotted.

Interpretation:

In the graph above, the black curve is the graph of $y=f\left(x\right)$ and the red curve is the graph of $y=f\left(x\right)+3$ .

Conclusion:

Therefore, we have sketched the graph of $y=f\left(x\right)+3$ .

c. $$

To determine

To graph:

To sketch the graph of $y=f\left(x+2\right)$ .

Answer to Problem 22E

The graph of $y=f\left(x+2\right)$ is given by

  

Explanation of Solution

Given information:

The given function is $y=f\left(x+2\right)$ and the given $f$ graph is

  

Graph:

  

Here, the $+2$ is grouped with the $x$ . Hence it is a horizontal translation.

Therefore, we transform the graph $f$ of by $2$ units to the left.

Calculation of Graph:

Consider $y=f\left(x+2\right)$ ,

Values of x	Values of y
$-4$	$4$
$-2$	$3$
$-1$	$0$
$1$	$-1$

The actual translation is to move the entire graph to the left two units or subtract the two from every x-coordinate. Using the above table, the graph can be plotted.

Interpretation:

In the graph above, the black curve is the graph of $y=f\left(x\right)$ and the red curve is the graph of $y=f\left(x+2\right)$ .

Conclusion:

Therefore, we have sketched the graph of $y=f\left(x+2\right)$ .

d. $$

To determine

To graph:

To sketch the graph of $y=f\left(x-1\right)$ .

Answer to Problem 22E

The graph of $y=f\left(x-1\right)$ is given by

  

Explanation of Solution

Given information:

The given function is $y=f\left(x-1\right)$ and the given $f$ graph is

  

Graph:

  

Here, we transform the graph $f\left(x\right)$ by $1$ units right since $-1$ is grouped with the $x$ , therefore it is a horizontal translation.

Calculation of Graph:

Consider $y=f\left(x-1\right)$

Values of x	Values of y
$-1$	$4$
$1$	$3$
$2$	$0$
$4$	$-1$

Using the above table, the graph can be plotted.

Interpretation:

In the graph above, the black curve is the graph of $y=f\left(x\right)$ and the red curve is the graph of $y=f\left(x-1\right)$ .

Conclusion:

Therefore, we have sketched the graph of $y=f\left(x-1\right)$ .

e. $$

To determine

To graph:

To sketch the graph of $y=-f\left(x-2\right)$ .

Answer to Problem 22E

The graph of $y=-f\left(x-2\right)$ is given by

  

Explanation of Solution

Given information:

The given function is $y=-f\left(x-2\right)$ and the given $f$ graph is

  

Graph:

  

Here, we transform the graph $f\left(x\right)$ by $2$ units to the right, which reflects across $x$ axis.

Calculation of Graph:

Consider $y=-f\left(x-2\right)$

Values of x	Values of y
$0$	$-4$
$2$	$-3$
$3$	$0$
$5$	$1$

Using the above table, the graph can be plotted.

Interpretation:

In the graph above, the black curve is the graph of $y=f\left(x\right)$ and the red curve is the graph of $y=-f\left(x-2\right)$ .

Conclusion:

Therefore, we have sketched the graph of $y=-f\left(x-2\right)$ .

f. $$

To determine

To graph:

To sketch the graph of $y=f\left(-x\right)$ .

Answer to Problem 22E

The graph of $y=f\left(-x\right)$ is given by

  

Explanation of Solution

Given information:

The given function is $y=f\left(-x\right)$ and the given $f$ graph is

  

Graph:

  

Here, we transform the graph $y=f\left(-x\right)$ reflects across $y$ axis.

Calculation of Graph:

Consider $y=f\left(-x\right)$

Values of x	Values of y
$2$	$4$
$0$	$3$
$-1$	$0$
$-3$	$-1$

In this transformation only the position of the graph in the coordinate plane is changed. The basic shape of graph remains unchanged. Using the above table, the graph can be plotted.

Interpretation:

In the graph above, the black curve is the graph of $y=f\left(x\right)$ and the red curve is the graph of $y=f\left(-x\right)$ .

Conclusion:

Therefore, we have sketched the graph of $y=f\left(-x\right)$ .

g. $$

To determine

To graph:

To sketch the graph of $y=\frac{1}{2}f\left(x\right)$ .

Answer to Problem 22E

The graph of $y=\frac{1}{2}f\left(x\right)$ is given by

  

Explanation of Solution

Given information:

The given function is $y=\frac{1}{2}f\left(x\right)$ and the given $f$ graph is

  

Graph:

  

Here, we transform the graph $y=\frac{1}{2}f\left(x\right)$ vertically compress by factor $\frac{1}{2}$ .

Calculation of Graph:

Consider $y=\frac{1}{2}f\left(x\right)$

Values of x	Values of y
$-2$	$2$
$0$	$1.5$
$1$	$0$
$3$	$-0.5$

Using the above table, the graph can be plotted.

Interpretation:

In the graph above, the black curve is the graph of $y=f\left(x\right)$ and the red curve is the graph of $y=\frac{1}{2}f\left(x\right)$ .

Conclusion:

Therefore, we have sketched the graph of $y=\frac{1}{2}f\left(x\right)$ .

h. $$

To determine

To graph:

To sketch the graph of $y=f\left(2x\right)$ .

Answer to Problem 22E

The graph of $y=f\left(2x\right)$ is given by

  

Explanation of Solution

Given information:

The given function is $y=f\left(2x\right)$ and the given $f$ graph is

  

Graph:

  

Here, the transformation of the graph $y=f\left(2x\right)$ occurs horizontally compressed by the factor $\frac{1}{2}$ .

Calculation of Graph:

Consider $y=f\left(2x\right)$

Values of x	Values of y
$-1$	$4$
$0$	$3$
$0.5$	$0$
$1.5$	$-1$

Using the above table, the graph can be plotted.

Interpretation:

In the graph above, the black curve is the graph of $y=f\left(x\right)$ and the red curve is the graph of $y=f\left(2x\right)$ .

Conclusion:

Therefore, we have sketched the graph of $y=f\left(2x\right)$ .