Chapter 4.7, Problem 47E

a.

To determine

Write an absolute value function that represents the path of the sunlight that reflects off the water.

Answer to Problem 47E

  $f(x)=2\left|x-3\right|$

Explanation of Solution

Given information:You are sitting on a boat on a lake .You can get a sunburn from the sunlight that hits you directly and also from the sunlight that reflects off the water.

  

Calculation:

The vertex of the path of the sunlight that reflects off the water is (3,0) so the function has the form $f(x)=a\left|x-3\right|+0\text{where}h=3\text{and}k=0.$ The slope to the right of the vertex is 2 so $a=2.$

So, the absolute function that represents the path of the sunlight that reflects off the water is $f(x)=2\left|x-3\right|$ .

b.

To determine

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

Answer to Problem 47E

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

Explanation of Solution

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

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 $a=2\text{and}h=3\text{and}k=0.$ So piecewise function is,

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

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

Piecewise function is,

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