Question

3 - 4a
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.

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.

1. for what values of ( m ) does this equation

apply?

1. find an equation for the oblique section of the graph.
2. for what values of ( m ) does the equation from question 14 apply?
3. 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: Analyze the horizontal section

The horizontal section has a constant \( y - \) value. From the point \((0,29)\) and \((300,29)\), the equation of a horizontal line is \( y = c \), where \( c \) is a constant. Here \( c = 29 \), so the equation is \( y=29 \).

Step2: Determine the domain of the horizontal - section

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

Step3: Find the equation of the oblique section

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

Step4: Determine the domain of the oblique - section

The oblique section starts at \( m = 300 \) (exclusive, because at \( m = 300 \) the function is defined by the horizontal line) and has no upper bound (assuming the trend continues). So the values of \( m \) for which \( y = 0.25m - 46 \) applies is \( m>300 \).

Step5: Analyze the basic rate and free minutes

The basic rate is the cost when \( m = 0 \), which is \( \$29 \). The free minutes are the number of minutes for which the cost does not increase from the basic rate, which is \( 300 \) minutes.

Step6: Calculate the cost per additional minute

The slope of the oblique line \( 0.25 \) represents the cost per additional minute.

Answer:

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