Chapter 1.4, Problem 19E

a.

To determine

To sketch a graph for $y=f(x)+2$ using the graph of $f.$

Explanation of Solution

Given:

  $y=f(x)+2$

  

Calculation:

Using the given graph,

change the value of $y$ as per the given condition $y=f(x)+2$

$x$	$y$	$y=f(x)+2$
0	-1	1
1	0	2
3	1	3
4	2	4

Plot the graph as in the table for $(x,f(x)+2)$

Graph:

  

Interpretation:

The graph for $y=f(x)+2$ using the graph of $f$ is shown as above.

b.

To determine

To sketch a graph for $y=-f(x)$ using the graph of $f.$

Explanation of Solution

Given:

  $y=-f(x)$

  

Calculation:

Using the given graph,

change the value of $y$ as per the given condition $y=-f(x)$

$x$	$y$	$y=-f(x)$
0	-1	1
1	0	0
3	1	-1
4	2	-2

Plot the graph as in the table for $(x,-f(x))$

Graph:

  

Interpretation:

The graph for $y=-f(x)$ using the graph of $f$ is shown as above.

c.

To determine

To sketch a graph for $y=f(x-2)$ using the graph of $f.$

Explanation of Solution

Given:

  $y=f(x-2)$

  

Calculation:

Using the given graph,

change the value of $y$ as per the given condition $y=f(x-2)$

$x$	$x-2$	$y$
0	2	-1
1	3	0
3	5	1
4	6	2

Plot the graph as in the table for $(x-2,y)$

Graph:

  

Interpretation:

The graph for $y=f(x-2)$ using the graph of $f$ is shown as above.

d.

To determine

To sketch a graph for $y=f(x+3)$ using the graph of $f.$

Explanation of Solution

Given:

  $y=f(x+3)$

  

Calculation:

Using the given graph,

change the value of $y$ as per the given condition $y=f(x+3)$

$x$	$y$	$y=f(x+3)$
0	-1	1
1	0	2
-3	1	-1
-2	2	0

Plot the graph as in the table for $(x,f(x+3))$

Graph:

  

Interpretation:

The graph for $y=f(x+3)$ using the graph of $f$ is shown as above.

e.

To determine

To sketch a graph for $y=2f(x)$ using the graph of $f.$

Explanation of Solution

Given:

  $y=2f(x)$

  

Calculation:

Using the given graph,

change the value of $y$ as per the given condition $y=2f(x)$

$x$	$y$	$y=2f(x)$
0	-1	-2
1	0	0
3	1	2
4	2	4

Plot the graph as in the table for $(x,y=2f(x))$

Graph:

  

Interpretation:

The graph for $y=2f(x)$ using the graph of $f$ is shown as above.

f.

To determine

To sketch a graph for $y=f(-x)$ using the graph of $f.$

Explanation of Solution

Given:

  $y=f(-x)$

  

Calculation:

Using the given graph,

change the value of $y$ as per the given condition $y=f(-x)$

$f(-x)$	$y$
0	-1
-1	0
-3	1
-4	2

Plot the graph as in the table for $(x,f(-x))$

Graph:

  

Interpretation:

The graph for $y=f(-x)$ using the graph of $f$ is shown as above.

g.

To determine

To sketch a graph for $y=f(\frac{1}{2}x)$ using the graph of $f.$

Explanation of Solution

Given:

  $y=f(\frac{1}{2}x)$

  

Calculation:

Let $g(x)=f(\frac{1}{2}x)$

Then,

  $g(0)=f(\frac{1}{2}(0))=f(0)=-1$

  $g(2)=f(\frac{1}{2}(2))=f(1)=0$

  $g(6)=f(\frac{1}{2}(6))=f(3)=1$

  $g(8)=f(\frac{1}{2}(8))=f(4)=2$

Graph:

  

Interpretation:

The graph for $y=f(\frac{1}{2}x)$ using the graph of $f$ is shown as above.