Chapter 1.6, Problem 44E

Solution

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

Given:

  $y=-3\sqrt{x+1}$

Graph:

  

Interpretation:

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

  $y=\sqrt{x}$

Assume that $y=\sqrt{x}$ is $f\left(x\right)=\sqrt{x}$ and $y=-3\sqrt{x+1}$ is $g\left(x\right)=-3\sqrt{x+1}$

  $f\left(x\right)=\sqrt{x}$

  $g\left(x\right)=-3\sqrt{x+1}$

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

  $y=a\sqrt{x-h}+k$

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

  $y=\sqrt{x}$

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

  $y=-3\sqrt{x+1}$

Find $a,h,$ and $k$ for $y=-3\sqrt{x+1}$

  $\begin{array}{l}a=-3\hfill \\ h=-1\hfill \\ k=0\hfill \end{array}$

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: $y=\sqrt{x}$

Horizontal Shift: Left $1$ Units

Vertical Shift: None

Reflection about the $x-$ axis: Reflected

Reflection about the $y-$ axis: None

Vertical Compression or Stretch: Stretched by factor of $3$