Chapter 1.7, Problem 27E

(a)

To determine

In Exercises 21−46, $g$ is related to one of the parent functions described in Section 1.6.

(a) Identify the parent function $f$

(b) Describe the sequence of transformations from $f$ to $g$

(c) Sketch the graph of $g$

(d) Use function notation to write $g$ in terms of $f$

27. $g\left(x\right)=2-{\left(x+5\right)}^2$

Identify the parent function $f$

Answer to Problem 27E

The Parent function is $f\left(x\right)=x^2$

Explanation of Solution

Given:

  $g\left(x\right)=2-{\left(x+5\right)}^2$

Concept Used:

A Parent function is the function in the simplest form possible for a given family of functions that tends to preserve the properties and definition of the entire family.

Conclusion:

The Parent function is $f\left(x\right)=x^2$

(b)

To determine

Describe the sequence of transformations from $f$ to $g$

Answer to Problem 27E

The shape of $f\left(x\right)=x^2$ , but shifted to the left by 5 units and reflected upon x-axis and the shifted upwards by 2 units.

Explanation of Solution

Given:

  $g\left(x\right)=2-{\left(x+5\right)}^2$

  $and$

  $f\left(x\right)=x^2$

Concept Used:

The sequence of transformations from $f$ to $g$ depicts the steps followed and the transformations used to reach from the Parent function $f$ to $g$ .

Conclusion:

The shape of $f\left(x\right)=x^2$ , but shifted to the left by 5 units and reflected upon x-axis and the shifted upwards by 2 units is the required sequence of transformations from $f$ to $g$ .

(c)

To determine

Sketch the graph of $g$ .

Answer to Problem 27E

  

Explanation of Solution

Given:

  $g\left(x\right)=2-{\left(x+5\right)}^2$

Concept Used:

We used the sequence of transformations to the plot the above graph.

Conclusion:

We thus obtain the graph for $g\left(x\right)=2-{\left(x+5\right)}^2$

(d)

To determine

Use function notation to write $g$ in terms of $f$

Answer to Problem 27E

  $g\left(x\right)=2-f\left(x+5\right)$

Explanation of Solution

Given:

  $g\left(x\right)=2-{\left(x+5\right)}^2$

  $and$

  $f\left(x\right)=x^2$

Conclusion:

  $g\left(x\right)=2-f\left(x+5\right)$