Question

joan plans to add 12 photos to her photo album each week. after 8 weeks, there are 120 photos in the album. assume the relationship is linear. find and interpret the rate of change and initial value. then write the equation of the function in the form y = mx + b, where x represents the number of weeks and y represents the total number of photos in the album. step 1: find and interpret the rate of change. step 2: find the initial value. step 3: write the equation of the function.

Explanation:

Step1: Identify rate of change

The rate of change is the number of photos added per week, so $m = 12$. This means the total number of photos in the album increases by 12 each week.

Step2: Calculate initial value $b$

Substitute $x=8$, $y=120$, $m=12$ into $y=mx+b$:
$$120 = 12(8) + b$$
$$120 = 96 + b$$
$$b = 120 - 96 = 24$$
This initial value is the number of photos in the album at week 0.

Step3: Write linear function

Substitute $m=12$ and $b=24$ into $y=mx+b$.

Answer:

Rate of change: 12, which means 12 photos are added to the album each week.
Initial value: 24, which means there were 24 photos in the album at the start (week 0).
Function equation: $y = 12x + 24$