Question

unit 4 summative ma.912.f.1.5
consider the data below which represents how much a cell phone plan charges per minute for a given number of minutes for 3 different plans.
plan a: table with minutes and cost
plan b: graph with points
plan c: ( f(x) = \frac{1}{5}x + 10 )

1. which plan has the highest rate of change?
2. which plan has the lowest y - intercept?
3. which plan costs the most for 70 minutes?

Explanation:

Step1: Analyze Plan A (Table)

For Plan A, let's find the rate of change (slope). Take two points, e.g., (0, 5) and (20, 7). The rate of change \( m_A=\frac{7 - 5}{20 - 0}=\frac{2}{20}=0.1 \) dollars per minute. The y - intercept (when \( x = 0 \)) is 5. For \( x = 70 \), cost \( C_A=5 + 0.1\times70=5 + 7 = 12 \).

Step2: Analyze Plan B (Graph)

From the graph, when \( x = 0 \), the y - intercept seems to be 0 (since it passes through the origin). Take two points, e.g., (20, 2) and (40, 4). Rate of change \( m_B=\frac{4 - 2}{40 - 20}=\frac{2}{20}=0.1 \)? Wait, no, maybe better points. Wait, the x - axis is minutes (0,20,40) and y - axis is cost (0,20,40,60? Wait, maybe misread. Wait, the graph: x - axis (minutes): 0,20,40; y - axis (cost): 0,20,40,60? No, maybe the graph has points. Wait, maybe Plan B: when x = 0, y = 0 (y - intercept 0). Let's take (20, 2) and (40, 4), so slope \( m_B=\frac{4 - 2}{40 - 20}=0.1 \)? Wait, no, maybe the y - axis is cost. Wait, maybe I misread. Alternatively, Plan C: \( f(x)=\frac{2}{5}x+15 \). So rate of change (slope) \( m_C=\frac{2}{5}=0.4 \) dollars per minute, y - intercept 15. For \( x = 70 \), cost \( C_C=\frac{2}{5}\times70 + 15=28 + 15 = 43 \).

Step3: Compare Rates of Change (Q1)

Plan A: \( m_A = 0.1 \), Plan B: Let's re - check. Wait, maybe Plan B's slope: if x = 20, y = 2; x = 40, y = 4? No, maybe the y - axis is different. Wait, Plan C has slope \( \frac{2}{5}=0.4 \), which is higher than Plan A's 0.1. So Plan C has the highest rate of change.

Step4: Compare Y - Intercepts (Q2)

Plan A: y - intercept 5; Plan B: y - intercept 0; Plan C: y - intercept 15. Wait, no, Plan B: if the graph passes through (0,0), y - intercept 0. Plan A: 5, Plan C:15. So Plan B has the lowest y - intercept.

Step5: Compare Costs for 70 minutes (Q3)

Plan A: \( C_A=5+0.1\times70 = 12 \); Plan B: Let's assume Plan B's slope: if x = 20, y = 2 (cost), then for x = 70, \( C_B=0.1\times70 = 7 \) (if slope 0.1); Plan C: \( C_C=\frac{2}{5}\times70+15 = 28 + 15 = 43 \). So Plan C costs the most.

Answer:

s:

1. Plan C (since its rate of change \( \frac{2}{5}=0.4 \) is highest)
2. Plan B (y - intercept 0, lowest)
3. Plan C (costs 43 for 70 minutes, highest)