Question

a shoe store offers free points when you sign up for their rewards card. then, for each pair of shoes purchased, you earn an additional number of points. the graph shows the total points earned for several pairs of shoes.
find and interpret the rate of change and initial value. then write the equation of the function in the form ( y = mx + b ).
graph of points earned vs. pairs of shoes purchased with a dashed line
step 1 find and interpret the rate of change.
step 2 find and interpret the initial value.
step 3 write the equation of the function in slope - intercept form.

Explanation:

Step1: Calculate rate of change

Choose two points from the graph, e.g., $(2, 60)$ and $(4, 90)$. Rate of change $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{90 - 60}{4 - 2} = \frac{30}{2} = 15$.
Interpretation: 15 points per shoe pair.

Step2: Find initial value $b$

Use point $(2,60)$ and $m=15$ in $y=mx+b$.
$60 = 15(2) + b$
$60 = 30 + b$
$b = 60 - 30 = 30$.
Interpretation: 30 sign-up points.

Step3: Write slope-intercept equation

Substitute $m=15$ and $b=30$ into $y=mx+b$.
$y = 15x + 30$

Answer:

1. Rate of change: 15, meaning you earn 15 additional points for each pair of shoes purchased.
2. Initial value: 30, meaning you earn 30 points when you sign up for the rewards card (0 pairs of shoes purchased).
3. Equation: $y = 15x + 30$