Chapter 1, Problem 75RE

a.

To determine

The parent function of f

Answer to Problem 75RE

Parent function will be $f\left(x\right)=x^2$

Explanation of Solution

Given:

The function is given as:

  $h\left(x\right)=-{\left(x+2\right)}^2-3$

Rule for shifting the function horizontally is;

  $h\left(x\right)=f\left(x+c\right)$ ( forward )

  $h\left(x\right)=f\left(x-c\right)$ ( backward )

Now, rule for shifting the function vertically is;

  $h\left(x\right)=f\left(x\right)+c$ ( upward)

  $h\left(x\right)=f\left(x\right)-c$ ( downward )

Here, function is given;

  $h\left(x\right)=-{\left(x+2\right)}^2-3$

Therefore parent function will be $f\left(x\right)=x^2$

b.

To determine

Sequence of transformation from f to h.

Answer to Problem 75RE

Transformation from f to h takes place by shifting $f\left(x\right)=x^2$ 2 units left then taking mirror image around x axis and then finally 3 units downward.

Explanation of Solution

Given:

The function is given as:

  $h\left(x\right)=-{\left(x+2\right)}^2-3$

Rule for shifting the function horizontally is;

  $h\left(x\right)=f\left(x+c\right)$ ( forward )

Here, c is real constant.

Now, rule for shifting the function vertically is;

  $h\left(x\right)=f\left(x\right)+c$ ( upward )

Here, c is real constant.

Function is given;

  $h\left(x\right)=-{\left(x+2\right)}^2-3$

It can be found by shifting $f\left(x\right)=x^2$ by 2 units left then taking mirror image around x axis and then finally 3 units downward.

c.

To determine

To sketch :

The graph of h.

Explanation of Solution

Given:

The function is given as:

  $h\left(x\right)=-{\left(x+2\right)}^2-3$

Plot of h is ;

d.

To determine

Function notation to write h in terms of f

Answer to Problem 75RE

  $h\left(x\right)=-f\left(x+2\right)-3$

Explanation of Solution

Given:

The function is given as:

  $h\left(x\right)=-{\left(x+2\right)}^2-3$

h in terms of f can be written as;

  $h\left(x\right)=-f\left(x+2\right)-3$