Chapter 1.6, Problem 46E

Solution

To Explain: The given basic graph and a sequence of transformation which used to produce given graph function.

Given:

  $y=-2\left|x+4\right|+1$

Graph:

  

Interpretation:

The parent function is the simplest form of the type of function given.

  $y=\left|x\right|$

Assume that $y=\left|x\right|$ is $f\left(x\right)=\left|x\right|$ and $y=-2\left|x+4\right|+1$ is $g\left(x\right)=-2\left|x+4\right|+1$

  $\begin{array}{c}f\left(x\right)=\left|x\right|\\ f\left(x\right)=-2\left|x+4\right|+1\end{array}$

The transformation from the first equation to the second one can be found by finding $a,h,$ and $k$ for each equation.

  $y=a\left|x-h\right|+k$

Factor a $1$ out of the absolute value to make the coefficient of $x$ equal to $1$ .

  $y=\left|x\right|$

Factor a $1$ out of the absolute value to make the coefficient of $x$ equal to $1$ .

  $y=-2\left|x+4\right|+1$

Find $a,h,$ and $k$ for $y=-2\left|x+4\right|+1$

  $\begin{array}{l}a=-2\hfill \\ h=-4\hfill \\ k=1\hfill \end{array}$

The horizontal shift depends on the value of  $h$ .

The vertical shift depends on the value of  $k$ .

The sign of  $a$  describes the reflection across the  $x-$ axis. 

  $-a$ means the graph is reflected across the  $x-$ axis.

Compare and list the transformations:

Parent Function: $y=\left|x\right|$

Horizontal Shift: Left $4$ units

Vertical Shift: Up $1$ units

Reflection about the $x-$ axis: Reflected

Reflection about the $y-$ axis: None

Vertical stretch: Stretched by a factor of $2$