Question

uses objective k

1. the table at the right shows the amount of federal income tax that a single person would pay at three of the 2007 tax rates. translate it to a piecewise linear function ( t(x) ) where ( x ) is the persons income.

t(x)=left{\begin{array}{l} \text{_} ; 0<x leq 7825 \\ \text{_} ; \text{_} <x leq \text{_} \\ \text{_} ; end{array}
ight.

1. a copy stores charges are shown at the right. rewrite the charges as a piecewise function ( c(n) ) where ( n ) is the number of copies.

representations objective n
in 12 - 16, refer to the graph at the right, which shows the cost of a cell phone plan as a function of the number of minutes ( m ) that are used in a given month.

1. find an equation for the horizontal section of the graph.
2. for what values of ( m ) does this equation apply?
3. find an equation for the oblique section of the graph.
4. for what values of ( m ) does the equation from question 14 apply?
5. fill in the blanks provide the details of the plan: the basic rate is _ per month including _ free minutes. additional minutes cost _ each.

Explanation:

Step1: Determine the equation for the horizontal section

The horizontal section of the graph has a constant \(y\) - value. Looking at the points \((0,29)\) and \((300,29)\), the equation of a horizontal line is \(y = c\), where \(c\) is a constant. So the equation is \(y=29\).

Step2: Find the values of \(m\) for the horizontal - section equation

The horizontal section of the graph starts at \(m = 0\) and ends at \(m = 300\). So the equation \(y = 29\) applies for \(0\leq m\leq300\).

Step3: Calculate the slope of the oblique section

The oblique section passes through the points \((300,29)\) and \((400,54)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1,y_1)=(300,29)\) and \((x_2,y_2)=(400,54)\). Then \(m=\frac{54 - 29}{400 - 300}=\frac{25}{100}=0.25\).
Using the point - slope form \(y - y_1=m(x - x_1)\), with \(m = 0.25\) and \((x_1,y_1)=(300,29)\), we have \(y-29 = 0.25(x - 300)\).
Expanding: \(y-29=0.25x-75\), so \(y = 0.25x- 46\).

Step4: Find the values of \(m\) for the oblique - section equation

The oblique section of the graph starts at \(m = 300\). So the equation \(y = 0.25x-46\) applies for \(m>300\).

Step5: Determine the basic rate and free minutes and additional - minute cost

The basic rate is the \(y\) - value when \(m = 0\), which is \(\$29\) per month. The number of free minutes is the \(x\) - value at the end of the horizontal section, which is \(300\) minutes.
The slope of the oblique section \(0.25\) (from \(y = 0.25x-46\)) represents the cost per additional minute. So additional minutes cost \(\$0.25\) each.

Answer:

1. \(y = 29\)
2. \(0\leq m\leq300\)
3. \(y=0.25x - 46\)
4. \(m>300\)
5. The basic rate is \(\$29\) per month including \(300\) free minutes. Additional minutes cost \(\$0.25\) each.