Question

a telephone company offers a monthly cellular phone plan for $35.00. it includes 350 free minutes plus $0.25 per minute for additional minutes. the following function gives the monthly cost for a subscriber, where x is the number of minutes used. simplify the expression in the second line of the piece - wise function. then use point - plotting to graph the function. $c(x)=\begin{cases}35.00 &\text{if }0leq xleq350\\35.00 + 0.25(x - 350)&\text{if }x>350end{cases}$ choose the correct graph of the function.

Explanation:

Step1: Simplify the second - part of the piece - wise function

\[

$$\begin{align*} C(x)&=35 + 0.25(x - 350)\\ &=35+0.25x-0.25\times350\\ &=35 + 0.25x-87.5\\ &=0.25x - 52.5 \end{align*}$$

\]

Step2: Analyze the first part of the piece - wise function

When \(0\leq x\leq350\), \(C(x) = 35\). This is a horizontal line segment on the graph with \(y = 35\) for \(x\) values from \(0\) to \(350\) (inclusive).

Step3: Analyze the second part of the piece - wise function

When \(x>350\), \(C(x)=0.25x - 52.5\). The slope of this line is \(m = 0.25=\frac{1}{4}\), and when \(x = 350\), \(C(350)=35\). We can find another point, for example, when \(x = 400\), \(C(400)=0.25\times400-52.5=100 - 52.5 = 47.5\).

The graph will be a horizontal line \(y = 35\) for \(0\leq x\leq350\) and then a line with a positive slope starting at the point \((350,35)\) for \(x>350\).

Answer:

The correct graph is the one that has a horizontal line at \(y = 35\) for \(0\leq x\leq350\) and then a line with a positive slope starting at the point \((350,35)\) for \(x>350\). Without seeing the exact details of the graphs A, B, C, D, based on the above - described characteristics, you can identify the correct one. If we assume the standard orientation of the \(x\) - axis as the number of minutes and \(y\) - axis as the cost, the graph should be flat at \(y = 35\) from \(x = 0\) to \(x = 350\) and then start increasing with a slope of \(0.25\) for \(x>350\).