Question

a telephone company offers a monthly cellular phone plan for $25.00. it includes 300 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) = {25.00 if 0 <= x < 300; 25.00 + 0.25(x - 300) if x > 300}
choose the correct graph of the function.

Explanation:

Step1: Simplify the second - part of the function

When \(x>300\), expand \(C(x)=25 + 0.25(x - 300)\). Using the distributive property \(a(b - c)=ab - ac\), we have \(C(x)=25+0.25x-0.25\times300=25 + 0.25x-75=0.25x - 50\).

Step2: Analyze the function for different intervals

For \(0\leq x<300\), \(C(x) = 25\), which is a horizontal line at \(y = 25\). For \(x>300\), \(C(x)=0.25x - 50\). When \(x = 300\), \(C(300)=25\). When \(x = 400\), \(C(400)=0.25\times400-50=100 - 50=50\).

Step3: Determine the graph

The graph has a horizontal line segment \(y = 25\) for \(0\leq x<300\) and then a line with slope \(m = 0.25\) starting at the point \((300,25)\) for \(x>300\).

Answer:

The correct graph is the one that has a horizontal line \(y = 25\) from \(x = 0\) to \(x = 300\) (excluding \(x = 300\) on the right - hand side of the horizontal part) and then a line with a positive slope starting at the point \((300,25)\) for \(x>300\). Without seeing the exact details of the options, based on the description, it should be a graph that has a flat part at \(y = 25\) for \(0\leq x<300\) and then an upward - sloping line for \(x>300\) starting at the point \((300,25)\).