Chapter 1.4, Problem 20E

a.

To determine

To sketch a graph for $y=f(x)-1$ using the graph of $f.$

Explanation of Solution

Given:

  $y=f(x)-1$

  

Calculation:

Using the given graph,

The given points are $(-2,4),(0,3),(1,0),(3,-1)$

There is a vertical shift one unit downward. Hence, subtract one from the $y$ axis.

So,

  $\begin{array}{l}(-2,4-1),(0,3-1),(1,0-1),(3,-1-1)\\ \Rightarrow (-2,3),(0,2),(1,-1),(3,-2)\end{array}$

Plot the above points in a graph.

Graph:

  

Interpretation:

The graph for $y=f(x)-1$ using the graph of $f$ is shown as above.

b.

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,

The given points are $(-2,4),(0,3),(1,0),(3,-1)$

There is a horizontal shift two units left. Hence, subtract two from the $x$ axis.

So,

  $\begin{array}{l}(-2-2,4),(0-2,3),(1-2,0),(3-2,-1)\\ \Rightarrow (-4,4),(-2,3),(-1,0),(1,-1)\end{array}$

Plot the above points in a graph.

Graph:

  

Interpretation:

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

c.

To determine

To sketch a graph for $y=f(x-1)$ using the graph of $f.$

Explanation of Solution

Given:

  $y=f(x-1)$

  

Calculation:

Using the given graph,

The given points are $(-2,4),(0,3),(1,0),(3,-1)$

There is a horizontal shift one unit right. Hence, add oneto the $x$ axis.

So,

  $\begin{array}{l}(-2+1,4),(0+1,3),(1+1,0),(3+1,-1)\\ \Rightarrow (-1,4),(1,3),(2,0),(4,-1)\end{array}$

Plot the above points in a graph.

Graph:

  

Interpretation:

The graph for $y=f(x-1)$ using the graph of $f$ is shown as above.

d.

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,

The given points are $(-2,4),(0,3),(1,0),(3,-1)$

There is a horizontal shift two units right. Hence, add two to the $x$ axis and find the negative of it.

So,

  $\begin{array}{l}(-2+2,4),(0+2,3),(1+2,0),(3+2,-1)\\ \Rightarrow (0,4),(-2,3),(-3,0),(-5,-1)\end{array}$

Plot the above points in a graph.

Graph:

  

Interpretation:

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

e.

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,

The given points are $(-2,4),(0,3),(1,0),(3,-1)$

Find the negative of the $x$ co-ordinate.

So,

  $\begin{array}{l}(-(-2),4),(0,3),(-(1),0),(-(3),-1)\\ \Rightarrow (2,4),(0,3),(-1,0),(-3,-1)\end{array}$

Plot the above points in a graph.

Graph:

  

Interpretation:

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

f.

To determine

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

Explanation of Solution

Given:

  $y=\frac{1}{2}f(x)$

  

Calculation:

Using the given graph,

Using the given graph,

The given points are $(-2,4),(0,3),(1,0),(3,-1)$

Find the $x$ co-ordinate divided and halved.

So,

  $\begin{array}{l}(\frac{1}{2}(-2),4),(\frac{1}{2}(0),3),(\frac{1}{2}(1),0),(\frac{1}{2}(3),-1)\\ \Rightarrow (-1,4),(0,3),(0.5,0),(1.5,-1)\end{array}$

Plot the above points in a graph.

Graph:

  

Interpretation:

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

g.

To determine

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

Explanation of Solution

Given:

  $y=f(2x)$

  

Calculation:

Using the given graph,

The given points are $(-2,4),(0,3),(1,0),(3,-1)$

Find twice of the $x$ co-ordinates.

So,

  $\begin{array}{l}(2\times (-2),4),(2\times 0,3),(2\times (1),0),(2\times (3),-1)\\ \Rightarrow (-4,4),(0,3),(2,0),(6,-1)\end{array}$

Plot the above points in a graph.

Graph:

  

Interpretation:

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