Chapter 1.7, Problem 37E

(a)

To determine

Find the parent function $f$ .

Answer to Problem 37E

For the function $g(x)=-\left|x+4\right|+8$ ,the parent function is $f(x)=\left|x\right|$ .

Explanation of Solution

Given information: $g(x)=-\left|x+4\right|+8$

Calculation:

Parent function is the basic function of a family of functions that preserves the definitions, shape of its graph and properties of the entire family.

Parent function used in this question is the absolute value function i.e$f(x)=\left|x\right|$

To identify the parent function, strip all the arithmetic operations on the function to leave behind one higher order operation in just x.

So, remove the arithmetic operation of multiplication by -1 and then addition of 4 to x and addition of 8 to it from the given funcion to get the parent function.

The parent function is $f(x)=\left|x\right|$

(b)

To determine

Find the sequence of transformation from f to g.

Answer to Problem 37E

The shape of $g(x)=-\left|x+4\right|+8$ is drawn reflected in the x-axis and then shifted left by 4 units and shifted upward by 8 units.

Explanation of Solution

Given information: $g(x)=-\left|x+4\right|+8$ and $f(x)=\left|x\right|$

Calculation:

The sequence of transformations from ƒ to g depicts the steps followed and the transformations used to reach from the parent function ƒ to g.

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

Types of shifts used in function transfromation:

Vertical shift: If $c$ is a real number which is also positive, thenthe graph of $f\left(x\right)+c$ is the graph of $y=f\left(x\right)$shifted upward by $c$ units.

If $c$ is a real number which is also positive, then the graph of $f\left(x\right)-c$ is the graph of $y=f\left(x\right)$shifted downwards by $c$ units.

Horizontal Shift:If $c$ is a real number which is also positive then, the graph of $f\left(x+c\right)$ is the graph of $y=f\left(x\right)$shifted left by $c$ units.

If $c$ is a real number which is also positive then, the graph of $f\left(x-c\right)$ is the graph of $y=f\left(x\right)$shifted right by $c$ units.

Reflection:The graph for the function say $y=f\left(-x\right)$ is the graph of $y=f\left(x\right)$is the reflection in y-axis.

The graph for the function say $y=-f\left(x\right)$ is the graph of $y=f\left(x\right)$ is the reflection in x-axis.

Vertical Stretching and Shrinking: If $c\succ 1$ then, the graph of $y=cf(x)$ is the graph of $y=f\left(x\right)$ stretched vertically by $c$ units.

If $0\prec c\prec 1$ then, the graph of $y=cf(x)$ is nothing but the graph of $y=f\left(x\right)$ shrunk vertically by $c$ units.

Horizontal Stretching and Shrinking: If $c\succ 1$ then, the graph of $y=cf(x)$ is nothing but the graph of $y=f\left(x\right)$ shrunk horizontally by $c$ units.

If $0\prec c\prec 1$ then, the graph of $y=cf(x)$ is nothing but the graph of $y=f\left(x\right)$ stretched horizontally by $c$ units.

Conclusion:

The shape of $g(x)=-\left|x+4\right|+8$ is drawn reflected in the x-axis and then shifted left by by 4 units and shifted upward by 8 units is the required sequence of transformations from ƒ to g.

(c)

To determine

To sketch the graph of $g$ .

Explanation of Solution

Given information: $g(x)=-\left|x+4\right|+8$

  $f(x)=\left|x\right|$

Calculation: Use the sequence of transformation to plot the graph of the function $g(x)=-\left|x+4\right|+8$

Obtain the graph of $g(x)=-\left|x+4\right|+8$

(d)

To determine

To write $g$ in terms of $f$

Answer to Problem 37E

  $g(x)=-f(x)+8$

Explanation of Solution

Given information: $g(x)=-\left|x+4\right|+8$ and $f(x)=\left|x\right|$

Calculation:

Use parent functionsand then move them around the coordinate plane through various types of shifts and thus write one function in terms of the other.

Multiply $f(x)$ by -1 and then add 8 to it to get $g(x)$ in terms of $f(x)$

Conclusion:

  $g(x)=-f(x)+8$