Question

janelle and xavier are training for a marathon. the equation and graph each represent the runners time in minutes, y, as a function of their distance in miles, x. janelle: y = 12x what is janelles speed? find the rate of change. janelles speed = 12 minutes per mile what is xaviers speed? find the rate of change. xaviers speed = minutes per mile

Explanation:

Step1: Recall the formula for rate of change (slope)

The rate of change (slope) for a linear function \(y = mx + b\) (or for a line passing through \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For a graph of \(y\) (time) vs \(x\) (distance), the slope \(m=\frac{\text{change in }y}{\text{change in }x}\), which represents time per mile.

Step2: Identify two points on Xavier's graph

From the graph, two points on Xavier's line are \((1,9)\) and \((2,18)\) (since when \(x = 1,y=9\) and when \(x = 2,y = 18\)).

Step3: Calculate the rate of change (slope)

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), substitute \(x_1 = 1,y_1=9,x_2 = 2,y_2 = 18\). Then \(m=\frac{18 - 9}{2 - 1}=\frac{9}{1}=9\).

Answer:

\(9\)