Question

1. a movie theater offers a reward program that charges a yearly membership fee and a discounted rate per movie ticket. the total cost for a program member to see 5 movies is $40 and the total cost for 12 movies is $75. assume the relation ship is linear, where x represents the number of movies and y represents the total cost. find and interpret the rate of change and initial value. (example 4)

Explanation:

Step1: Find the slope (rate of change)

The rate of change (slope \(m\)) of a linear equation \(y = mx + b\) is calculated using the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(5,40)\) and \((x_2,y_2)=(12,75)\). Then \(m=\frac{75 - 40}{12 - 5}=\frac{35}{7} = 5\).

Step2: Find the initial value (y - intercept)

We use the point - slope form \(y - y_1=m(x - x_1)\). Using \(m = 5\) and the point \((5,40)\), we have \(y-40 = 5(x - 5)\). Expand: \(y-40=5x-25\). Then \(y=5x + 15\). The initial value (when \(x = 0\)) is found by substituting \(x = 0\) into the equation \(y=5x + 15\), so \(y=15\).

Answer:

The rate of change is \(5\) (meaning each movie ticket costs \(5\) dollars more in total cost when considering the membership) and the initial value is \(15\) (the cost of the yearly membership fee).