Question

janelle and xavier are training for a marathon. the equation and graph each represent the runner’s time in minutes, y, as a function of their distance in miles, x. janelle: y = 12x what is janelle’s speed? find the rate of change. janelle’s speed = 12 minutes per mile what is xavier’s speed? find the rate of change. xavier’s speed = 6 minutes per mile rate of change = (change in y - values)/(change in x - values)

Explanation:

Step1: Recall rate of change formula

The rate of change (slope) for a linear function \( y = mx + b \) is \( m \), and for a graph, it's \( \frac{\text{change in } y}{\text{change in } x} \).

Step2: Janelle's speed (rate of change)

Janelle's equation is \( y = 12x \). Here, the slope \( m = 12 \), so her rate of change (speed) is \( 12 \) minutes per mile.

Step3: Xavier's speed (rate of change)

From the graph, pick two points. At \( x = 2 \), \( y = 12 \) (wait, no, looking at the graph: when \( x = 2 \), \( y = 12 \)? Wait, no, the graph: when \( x = 1 \), \( y = 6 \)? Wait, no, the graph has \( x \) as distance (mi) and \( y \) as time (min). Let's take two points: (0,0) and (2,12)? Wait, no, the line goes from (0,0) to (2, 12)? Wait, no, the y-axis is time (min), x-axis distance (mi). Wait, the formula is \( \frac{\text{change in } y}{\text{change in } x} \). Let's take \( x = 2 \), \( y = 12 \)? Wait, no, the graph: when \( x = 1 \), what's \( y \)? Wait, the grid: each x-grid is 1 mile, y-grid is 2 minutes? Wait, no, the y-axis is 0,2,4,6,8,10,12,14,16,18. x-axis 0,1,2,3,... So when \( x = 2 \), \( y = 12 \)? Wait, no, the line goes from (0,0) to (2, 12)? Wait, no, the line is steep. Wait, the formula for rate of change is \( \frac{\Delta y}{\Delta x} \). Let's take two points: (0,0) and (2, 12)? Wait, no, if \( x = 2 \), \( y = 12 \), then \( \frac{12 - 0}{2 - 0} = 6 \)? Wait, no, maybe I misread. Wait, the problem says Xavier's speed is 6 minutes per mile. Let's check: if \( x = 2 \) miles, time \( y = 12 \) minutes? No, wait the graph: when \( x = 1 \), \( y = 6 \)? Wait, the y-axis at x=1: the line is at y=6? Wait, no, the graph: the line starts at (0,0), goes up. Let's take x=2, y=12? No, that would be 6 min per mile. Wait, \( \frac{12 - 0}{2 - 0} = 6 \) minutes per mile. Yes, so \( \frac{\Delta y}{\Delta x} = \frac{12}{2} = 6 \) minutes per mile.

Answer:

Janelle's speed: 12 minutes per mile, Xavier's speed: 6 minutes per mile. (But the question was about finding Xavier's speed, and the answer is 6 minutes per mile, and Janelle's is 12. But the problem's Xavier's speed was to be found, and the calculation shows \( \frac{\text{change in } y}{\text{change in } x} = \frac{12 - 0}{2 - 0} = 6 \) minutes per mile.)