Question

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?
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.

Explanation:

Step1: Define cost functions

For the prepaid plan, cost \( y_1 = 0.20x \) (slope \( m_1 = 0.20 \), y - intercept \( b_1 = 0 \)).
For the contracted plan, cost \( y_2 = 0.02x + 50 \) (slope \( m_2 = 0.02 \), y - intercept \( b_2 = 50 \)).

Step2: Compare slopes (steepness)

Slope represents rate of change. \( 0.20>0.02 \), so prepaid plan has steeper slope (steeper line).

Step3: Compare y - intercepts

Y - intercept is the cost when \( x = 0 \) (no minutes used). Prepaid: \( b_1 = 0 \); Contracted: \( b_2 = 50 \). So prepaid has lower y - intercept.

Answer:

The prepaid phone plan will have a steeper line and lower y - intercept.