Chapter 1, Problem 14PS

(a)

To determine

To Sketch: The sketch for the graph of the function $f\left(x+1\right)$ .

Answer to Problem 14PS

The required sketch for the graph is shown in Figure 2

Explanation of Solution

Given:

The given function is $f\left(x+1\right)$ .

The given graph is shown in Figure 1

  

Figure 1

Calculation:

The above function is the graph of the quadratic function, the general form of the quadratic function is,

  $f\left(x\right)=a{x}^2+bx+c$

To obtain the value of $c$ substitute 0 for $x$ in the above equation.

  $\begin{array}{l}f\left(0\right)=a{\left(0\right)}^2+b\left(0\right)+c\\ 2=c\end{array}$

Substitute 2 for $c$ in the general equation.

  $f\left(x\right)=a{x}^2+bx+2$ ..............(I)

Substitute $1$ for $x$ in the above equation.

  $\begin{array}{l}f\left(1\right)=a{\left(1\right)}^2+b\left(1\right)+2\\ 0=a+b+2\\ a+b=-2\end{array}$ ..............(II)

Substitute $-1$ for $x$ in equation (I)

  $\begin{array}{l}f\left(-1\right)=a{\left(-1\right)}^2+b\left(-1\right)+2\\ 0=a-b+2\\ a-b=-2\end{array}$

From equation (1) and from above,

  $\begin{array}{l}a-b+a+b=-2-2\\ 2a=-4\\ a=-2\end{array}$

Then,

  $\begin{array}{l}a+b=-2\\ -2+b=-2\\ b=0\end{array}$

Substitute $0$ for $b$ and $-2$ for $a$ in the general form of the quadratic equation.

  $\begin{array}{c}f\left(x\right)=-2x^2+\left(0\right)x+2\\ =-2x^2+2\end{array}$ ..............(III)

The function $f\left(x+1\right)$ is obtained by substituting $x+1$ for $x$ in the equation (III).

  $\begin{array}{c}f\left(x+1\right)=-2{\left(x+1\right)}^2+2\\ =-2x^2+4x\end{array}$

The sketch for the function is shown in Figure 2

  

Figure 2

(b)

To determine

To Sketch: The sketch for the graph of the function $f\left(x\right)+1$ .

Answer to Problem 14PS

The required sketch for the graph is shown in Figure 3

Explanation of Solution

The function $f\left(x\right)+1$ is obtained by adding1 to the equation (III).

  $\begin{array}{c}f\left(x\right)+1=-2{\left(x\right)}^2+2+1\\ =-2x^2+3\end{array}$

The sketch for the function is shown in Figure 3

  

Figure 3

(c)

To determine

To Sketch: The sketch for the graph of the function $2f\left(x\right)$ .

Answer to Problem 14PS

The required sketch for the graph is shown in Figure 4

Explanation of Solution

The function $2f\left(x\right)$ is obtained by multiplying equation (III) by 2.

  $\begin{array}{c}2f\left(x\right)=2\left(-2{\left(x\right)}^2+2\right)\\ =-4x^2+4\end{array}$

The sketch for the function is shown in Figure 4

  

Figure 4

(d)

To determine

To Sketch: The sketch for the graph of the function $f\left(-x\right)$ .

Answer to Problem 14PS

The required sketch for the graph is shown in Figure 5

Explanation of Solution

The function $f\left(-x\right)$ is the same as the graph $f\left(x\right)$ .

  $\begin{array}{c}f\left(-x\right)=-2{\left(-x\right)}^2+2\\ =-2{\left(x\right)}^2+2\end{array}$

The sketch for the function is shown in Figure 5

  

Figure 5

(e)

To determine

To Sketch: The sketch for the graph of the function $-f\left(x\right)$ .

Answer to Problem 14PS

The required sketch for the graph is shown in Figure 6

Explanation of Solution

The function $-f\left(x\right)$ is the mirror image of thegraph $f\left(x\right)$ .

  $\begin{array}{c}-f\left(x\right)=-\left(-2{\left(x\right)}^2+2\right)\\ =2{\left(x\right)}^2-2\end{array}$

The sketch for the function is shown in Figure 6

  

Figure 6

(f)

To determine

To Sketch: The sketch for the graph of the function $\left|f\left(x\right)\right|=\left|-2x^2+2\right|$ .

Answer to Problem 14PS

The required sketch for the graph is shown in Figure 7

Explanation of Solution

The function $\left|f\left(x\right)\right|=\left|-2x^2+2\right|$ will have all the negative values get reflected from the negative x axis.

The sketch for the function is shown in Figure 7

  

Figure 7

(g)

To determine

To Sketch: The sketch for the graph of the function $f\left|\left(x\right)\right|=-2{\left|x\right|}^2+2$ .

Answer to Problem 14PS

The required sketch for the graph is shown in Figure 8

Explanation of Solution

The absolute value of the function $f\left|\left(x\right)\right|=-2{\left|x\right|}^2+2$ does not matter as the value of $x$ is squared therefore the function will have no change from the function $f\left(x\right)$ .

The sketch for the function is shown in Figure 8

  

Figure 8