Chapter 2.7, Problem 34PPSE

Solution

(a).

To draw: The graph of the given equations in same coordinates.

Given information:

The given equations are $y=2\left|x+1\right|,y=\left|x+1\right|$ .

Calculation:

The given equation is $y=2\left|x+1\right|$ .

Compare $y=4\left|x-3\right|+1$ with the general form $y=a\left|x-h\right|+k$ :

  $a=2,h=-1,k=0$

The vertex $\left(h,k\right)$ is $\left(-1,0\right)$ .

The axis of symmetry is $x=-1$ .

The parent function $y=\left|x\right|$ is translated $1$ unit to the left, and vertically stretched by the factor $2$ .

Now, draw the graph of the function $y=2\left|x+1\right|$ is below:

  

The given equation is $y=\left|x+1\right|$ .

Compare $y=4\left|x-3\right|+1$ with the general form $y=a\left|x-h\right|+k$ :

  $a=1,h=-1,k=0$

The vertex $\left(h,k\right)$ is $\left(-1,0\right)$ .

The axis of symmetry is $x=-1$ .

The parent function $y=\left|x\right|$ is translated $1$ unit to the left.

Now, draw the graph of the function $y=\left|x+1\right|$ is below:

  

Now, draw the graph of both equations in the same coordinates are below:

  

(b).

To draw: The graph of the given equations in same coordinates.

Given information:

The given equations are $y=5\left|x-2\right|,y=\left|5x-2\right|$ .

Calculation:

The given equation is $y=5\left|x-2\right|$ .

Compare $y=5\left|x-2\right|$ with the general form $y=a\left|x-h\right|+k$ :

  $a=5,h=2,k=0$

The vertex $\left(h,k\right)$ is $\left(2,0\right)$ .

The axis of symmetry is $x=2$ .

The parent function $y=\left|x\right|$ is translated $2$ unit to the right, and vertically stretched by the factor $5$ .

Now, draw the graph of the function $y=5\left|x-2\right|$ is below:

  

The given equation is $y=\left|5x-2\right|$ .

Rewrite the above equation:

  $\begin{array}{c}y=\left|5\left(x-\frac{2}{5}\right)\right|\\ y=5\left|x-\frac{2}{5}\right|\end{array}$

Compare $y=5\left|x-\frac{2}{5}\right|$ with the general form $y=a\left|x-h\right|+k$ :

  $a=5,h=\frac{2}{5},k=0$

The vertex $\left(h,k\right)$ is $\left(\frac{2}{5},0\right)$ .

The axis of symmetry is $x=\frac{2}{5}$ .

The parent function $y=\left|x\right|$ is translated $\frac{2}{5}$ unit to the right, and vertically stretched by the factor $5$ .

  

Now, draw the graph of both equations in the same coordinates are below:

  

(c).

To compare: The graphs of the both equations in the part (a) and part (b).

The graph of function $y=\left|x+1\right|$ is translated vertically stretched by the factor $2$ of the graph of the function $y=\left|x+1\right|$ .

The graph of function $y=\left|5x-2\right|$ is translated $2$ unit to the right of the graph of the function $y=5\left|x-2\right|$ .

Given information:

Part (a):

The given equations are $y=2\left|x+1\right|,y=\left|x+1\right|$ .

Part (b):

The given equations are $y=5\left|x-2\right|,y=\left|5x-2\right|$ .

Calculation:

Part (a):

The given equations are $y=2\left|x+1\right|,y=\left|x+1\right|$ .

The graph of function $y=\left|x+1\right|$ is translated vertically stretched by the factor $2$ of the graph of the function $y=\left|x+1\right|$ .

Part (b):

The given equations are $y=5\left|x-2\right|,y=\left|5x-2\right|$ .

The graph of function $y=\left|5x-2\right|$ is translated $2$ unit to the right of the graph of the function $y=5\left|x-2\right|$ .