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 z represents minutes purchased in a month?
the contracted phone plan will have a steeper line and same 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 prepaid phone plan will have a steeper line and lower y - intercept.

Explanation:

Step1: Write the equations for both plans

The equation for a line in slope - intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.
For the contracted phone plan: The monthly fee \(b = 50\) (the \(y\) - intercept when \(x = 0\) minutes) and the cost per minute \(m=0.02\). So the equation is \(y = 0.02x+50\).
For the prepaid phone plan: The monthly fee \(b = 0\) (since there is no monthly fee) and the cost per minute \(m = 0.20\). So the equation is \(y=0.20x+0\).

Step2: Analyze the slopes and \(y\) - intercepts

• Slope (\(m\)): The slope represents the rate of change (cost per minute). A larger value of \(m\) means a steeper line. Since \(0.20>0.02\), the prepaid plan has a steeper slope.
• \(y\) - intercept (\(b\)): The \(y\) - intercept represents the fixed cost (monthly fee when \(x = 0\) minutes). The contracted plan has \(b = 50\) and the prepaid plan has \(b = 0\). So the contracted plan has a higher \(y\) - intercept.

Answer:

The contracted phone plan will have a steeper line and a higher \(y\) - intercept.