Question

a telephone company offers a monthly cellular phone plan for $20.00. it includes 350 free minutes plus $0.20 per minute for additional minutes. the following function gives the monthly cost to a subscriber, where x is the number of minutes used. simplify the expression in the second line of the piecewise function. then use point-plotting to graph the function.
c(x) =
\

$$\begin{cases} 20.00 & \\text{if } 0 \\leq x \\leq 350 \\\\ 20.00 + 0.20(x - 350) & \\text{if } x > 350 \\end{cases}$$

c(x) =
\

$$\begin{cases} 20.00 & \\text{if } 0 \\leq x \\leq 350 \\\\ 0.20x - 50.00 & \\text{if } x > 350 \\end{cases}$$

(use integers or decimals for any numbers in the expression. do not factor.)

choose the correct graph of the function.

\bigcirc a. \quad \bigcirc b. \quad \bigcirc c. \quad \bigcirc d.
(graphs for a, b, c, d are shown with axes and gridlines.)

Explanation:

Step1: Simplify the piecewise function's second part

We start with the expression \( 20.00 + 0.20(x - 350) \). First, distribute the \( 0.20 \) to both terms inside the parentheses:
\( 20.00 + 0.20x - 0.20\times350 \)

Step2: Calculate the constant term

Calculate \( 0.20\times350 \). \( 0.20\times350 = 70.00 \). Now substitute this back into the expression:
\( 20.00 + 0.20x - 70.00 \)

Step3: Combine like terms

Combine the constant terms \( 20.00 - 70.00 \):
\( 0.20x - 50.00 \)

For the graph:

• For \( 0\leq x\leq350 \), the cost \( C(x) = 20.00 \), so this is a horizontal line segment from \( x = 0 \) to \( x = 350 \) at \( y = 20 \).
• For \( x > 350 \), the cost is \( C(x)=0.20x - 50 \), which is a linear function with a slope of \( 0.20 \). To check the value at \( x = 350 \), substitute \( x = 350 \) into \( 0.20x - 50 \): \( 0.20\times350 - 50 = 70 - 50 = 20 \), which matches the first part at \( x = 350 \), so the graph should have a horizontal line until \( x = 350 \) and then a line with positive slope after that. Looking at the options, option B has a horizontal segment until \( x = 350 \) and then a steeper line (positive slope) after, which matches our function.

Answer:

The simplified expression for \( x > 350 \) is \( 0.20x - 50.00 \), and the correct graph is Option B.