Chapter 1.7, Problem 39E

(a)

To determine

Find the parent function $f$ .

Answer to Problem 39E

For the function $g(x)=-2\left|x-1\right|-4$ ,the parent function is $f(x)=\left|x\right|$ .

Explanation of Solution

Given information: $g(x)=-2\left|x-1\right|-4$

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.

Conclusion:

So,

remove the arithmetic operation of subtractionof 1 from x and multiplication by -2 and then subtraction of o $g$

  $g(x)=-2\left|x-1\right|-4$

f 4 from the given funcion to get the parent function.

(b)

To determine

Find the sequence of transformation from f to g.

Answer to Problem 39E

The shape of is drawn reflected in the x-axis and then shifted right by 1 unit and stretched by 2 units and then shifted downward by 4 units is the required sequence of transformations from $f$ to g.

Explanation of Solution

Given information: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:

1. 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.

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.

The shape of $g(x)=-2\left|x-1\right|-4$ is drawn reflected in the x-axis and then shifted right by 1 unit and stretched by 2 units and then shifted downward by 4 units.

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.

Conclusion:

The shape of is drawn reflected in the x-axis and then shifted right by 1 unit and stretched by 2 units and then shifted downward by 4 units is the required sequence of transformations from $f$ to $g$ .

(c)

To determine

To sketch the graph of $g$ .

Answer to Problem 39E

  

Explanation of Solution

Given information: $g(x)=-2\left|x-1\right|-4$

  $f(x)=\left|x\right|$

Calculation: Use the sequence of transformation to plot the graph of the function

Obtain the graph of $g(x)=-2\left|x-1\right|-4$

(d)

To determine

To write $g$ in terms of $f$

Answer to Problem 39E

  $g(x)=-2f(x)-4$

Explanation of Solution

Given information: $g(x)=-2\left|x-1\right|-4$ and $f(x)=\left|x\right|$

Calculation:

Multiply $f(x)$ by -2 and then subtract from from the parent functionto get $g(x)$ in terms of $f(x)$