Chapter 1.7, Problem 40E

(a)

To determine

Find the parent function $f$ .

Answer to Problem 40E

For the function $g(x)=\frac{1}{2}\left|x-2\right|-3$ ,the parent function is $f(x)=\left|x\right|$ .

Explanation of Solution

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.

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

Conclusion:

So, remove the arithmetic operation of subtraction of 2 from 2 and multiplication by 1/2 and then subtraction of 3 from the given funcion to get the parent function.

(b)

To determine

Find the sequence of transformation from f to g.

Answer to Problem 40E

The shape of $g(x)=\frac{1}{2}\left|x-2\right|-3$ is drawn reflected in the x-axis and stretched by by 1/2 units and then shifted downward by 3 units.

Explanation of Solution

Given information: $g(x)=\frac{1}{2}\left|x-2\right|-3$ and $f(x)=x^3$

Concept Used:

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:

1. Vertical shift: 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 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.

2. 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.

3. 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.

4. 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.

5. 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 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 shape of drawn reflected in the x-axis and stretched by by 1/2 units and then shifted downward by 3 units is the required sequence of transformations from ƒ to g.

(c)

To determine

To sketch the graph of $g$ .

Answer to Problem 40E

  $g(x)=\frac{1}{2}f(x)-3$

Explanation of Solution

Given information: $f(x)=\left|x\right|$

  $g(x)=\frac{1}{2}\left|x-2\right|-3$

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.

Conclusion:

Obtain the graph of $g(x)=\frac{1}{2}\left|x-2\right|-3$

(d)

To determine

To write $g$ in terms of $f$

Answer to Problem 40E

  $g(x)=\frac{1}{2}f(x)-3$

Explanation of Solution

Given information: $g(x)=\frac{1}{2}\left|x-2\right|-3$ and $f(x)=\left|x\right|$

Multiply $f(x)$ by ½ and then subtract 3 from it to get $g(x)$ in terms of $f(x)$ .