Question

1. else has a gift card to a local movie theater. the graph shows the amount of money remaining on her gift card based on the number of movies she has seen.

a. write an equation to represent the situation.
b. what is the y - intercept, and what does it mean in the context of the situation?
c. what is the slope, and what does it mean in the context of the situation?
the slope is - 8. it means the gift - card balance decreases by 8 dollars per movie seen.

Explanation:

Step1: Find the slope - intercept form of a linear equation

The slope - intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.

Step2: Determine the slope (\(m\))

We can use the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's assume two points from the context. If we consider the initial balance (when \(x = 0\)) and after some movies. The general form of the change in balance: If the balance changes by a certain amount per movie. From the equation \(y=-8x + 50\) (derived from the slope - intercept form and the context of cost per movie and initial balance), the slope \(m=-8\).

Step3: Determine the \(y\) - intercept (\(b\))

When \(x = 0\) (no movies seen), \(y=b\). In the context of the gift - card balance, when \(x = 0\) (she has seen \(0\) movies), \(y = 50\), so \(b = 50\).

Step4: Interpret the slope in context

The slope \(m=-8\). In the context of the gift - card balance, \(x\) is the number of movies seen and \(y\) is the balance. So, the slope means that the balance on the gift card decreases by \(8\) dollars per movie seen.

Step5: Interpret the \(y\) - intercept in context

The \(y\) - intercept \(b = 50\). When \(x = 0\) (number of movies seen is \(0\)), \(y=50\). So, it means the initial balance on the gift card is \(50\) dollars.

Answer:

a. The equation is \(y=-8x + 50\) (where \(y\) is the gift - card balance and \(x\) is the number of movies seen).
b. The \(y\) - intercept is \(50\), which means the initial balance on the gift card is \(50\) dollars.
c. The slope is \(-8\), which means the balance on the gift card decreases by \(8\) dollars per movie seen.