Grace is comparing cell phone plans. A prepaid phone plan costs $0.20 per minute and has no monthly fee. A contracted phone plan costs $50 per month and $0.02 per minute. How will the graphs of the monthly costs of the two cell phone plans compare where x represents minutes purchased in a month?

Question

The prepaid phone plan will have a steeper line and lower y-intercept.
The contracted phone plan will have the same steepness and a higher y-intercept.
The prepaid phone plan will have a less steep line and the same y-intercept.
The contracted phone plan will have a steeper line and same y-intercept.

Answer 1

To compare the two cell phone plans graphically, let's find the equations for the cost of both plans in terms of the number of minutes purchased, \( x \).

1. Prepaid Phone Plan:

   • Cost = $0.20 x (where x is the number of minutes)
   • Equation: \( C_{\text{prepaid}} = 0.20x \)

2. Contracted Phone Plan:

   • Cost = $50 (monthly fee) + $0.02 per minute
   • Equation: \( C_{\text{contracted}} = 50 + 0.02x \)

Now we can analyze the characteristics of each equation:

• The slope of the prepaid plan is \( 0.20 \).
• The slope of the contracted plan is \( 0.02 \).
• The y-intercept of the prepaid plan is \( 0 \) (since there is no monthly fee).
• The y-intercept of the contracted plan is \( 50 \) (due to the monthly fee).

Comparison:

• Steepness: The prepaid plan has a steeper slope (0.20 vs. 0.02), which means its cost increases more rapidly with more minutes than the contracted plan.
• Y-Intercept: The contracted plan has a higher y-intercept ($50 vs. $0), indicating that it has a base cost because of the monthly fee.

Given this information, we can determine how the graphs will compare:

• The prepaid phone plan will have a steeper line (higher slope) and a lower y-intercept (0).
• The contracted phone plan will have a less steep line and a higher y-intercept (50).

Thus, the correct statement about their comparison would be: The prepaid phone plan will have a steeper line and lower y-intercept.