Chapter 4.7, Problem 48E

a.

To determine

Write an absolute value function that represents the path of the golf ball.

Answer to Problem 48E

  $f(x)=-(2/3)\left|x-6\right|+4.$

Explanation of Solution

Given information: You are trying to make a hole in one on the miniature golf green.

  

Calculation:

The vertex of the path of the golf ball is (6,4) so the function has the form $f(x)=a\left|x-6\right|+4\text{where}h=6\text{and}k=4.$ Substitutes the co-ordinates of points (0,0) into the equation and solve for a .

  $\begin{array}{c}f(x)=a\left|x-6\right|+4.\\ 0=a\left|0-6\right|+4.\\ 0=6a+4.\\ 6a=-4.\\ a=-4/6.\\ a=-2/3.\end{array}$

So, the absolute function that represents the path of golf ball is $f(x)=-(2/3)\left|x-6\right|+4.$

b.

To determine

Write the function in part (a) as a piecewise function .

Answer to Problem 48E

  $\begin{array}{l}f(x)=(2/3)x,\text{if}x<6\text{and}\\ f(x)=(-2/3)x+8,\text{if}x\ge 6.\end{array}$

Explanation of Solution

Given information: Given absolute function is $f(x)=-(2/3)\left|x-6\right|+4.$

Calculation:

Formula:

Absolute value function $f(x)=a\left|x-h\right|+k$ can be written as piecewise function as,

  $\begin{array}{l}f(x)=a\left[-(x-h)\right]+k\text{if}x-h<0\text{and}\\ f(x)=a(x-h)+k\text{if}x-h\ge 0.\end{array}$

Here in given absolute function $a=(-2/3)\text{and}h=6\text{and}k=4.$ So piecewise function is,

  $\begin{array}{l}f(x)=(-2/3)\left[-(x-6)\right]+4\text{if}x-6<0\text{and}\\ f(x)=(-2/3)\left[x-6\right]+4\text{if}x-6\ge 0.\\ f(x)=(-2/3)\left[-x+6\right]\text{if}x<6\text{and}\\ f(x)=(-2/3)\left[x-6\right]\text{if}x\ge 6.\\ f(x)=(2/3)x\text{if}x<6\text{and}\\ f(x)=(-2/3)x+8\text{if}x\ge 6.\end{array}$

So from absolute function $f(x)=2\left|x-3\right|.$

Piecewise function is ,

  $\begin{array}{l}f(x)=(2/3)x,\text{if}x<6\text{and}\\ f(x)=(-2/3)x+8,\text{if}x\ge 6.\end{array}$