Question

The table shows the amount of pet food in cups remaining in an automatic feeder as a function of the number of meals the feeder has dispensed.

Number of meals dispenced x 1, 3 ,6 ,7

number of pet food remaining Y 21, 15, 6, 3

Write a function rule that models this situation:

Answer 1

To find a function rule that models the relationship between the number of meals dispensed \( x \) and the amount of pet food remaining \( y \), we can start by examining the data provided.

The data points are:

- When \( x = 1 \), \( y = 21 \)
- When \( x = 3 \), \( y = 15 \)
- When \( x = 6 \), \( y = 6 \)
- When \( x = 7 \), \( y = 3 \)

First, we can look at the change in the amount of food remaining as the number of meals dispensed increases.

1. From \( x = 1 \) to \( x = 3 \):
\[
21 - 15 = 6 \text{ cups over } 2 \text{ meals} \Rightarrow 3 \text{ cups per meal}
\]

2. From \( x = 3 \) to \( x = 6 \):
\[
15 - 6 = 9 \text{ cups over } 3 \text{ meals} \Rightarrow 3 \text{ cups per meal}
\]

3. From \( x = 6 \) to \( x = 7 \):
\[
6 - 3 = 3 \text{ cups over } 1 \text{ meal} \Rightarrow 3 \text{ cups per meal}
\]

The amount of food decreases uniformly by 3 cups for every meal dispensed.

Now, we can represent the relationship with a linear function. The general form of a linear equation is:
\[
y = mx + b
\]
where:
- \( m \) is the slope (change in \( y \) per unit change in \( x \))
- \( b \) is the \( y \)-intercept (the value of \( y \) when \( x = 0 \))

From our calculations, the slope \( m = -3 \) (since the amount of food decreases).

Next, we can calculate the \( y \)-intercept by plugging in one of the points. Let's use the point \( (1, 21) \):
\[
y = -3x + b
\]
Substituting \( x = 1 \) and \( y = 21 \):
\[
21 = -3(1) + b
\]
\[
21 = -3 + b
\]
\[
b = 21 + 3 = 24
\]

Thus, the function rule that models this situation is:
\[
y = -3x + 24
\]

This function indicates that for every meal dispensed, the amount of pet food decreases by 3 cups, starting from a total of 24 cups when no meals have been dispensed.