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.

Write a function rule, in slope-intercept form, that models this situation:

y=

Number of Meals Dispersed, x
Amount of Pet Food Remaining,
y
1
5
25
3
17
9
7
1

Answer 1

To model the relationship between the number of meals dispensed (x) and the amount of pet food remaining (y) using a linear function in slope-intercept form \( y = mx + b \), we need to determine the slope (m) and the y-intercept (b) based on the data you've provided.

From your data, we have the following points:

- Meal 1: (1, 25)
- Meal 2: (3, 17)
- Meal 3: (9, 7)

1. **Calculate the Slope (m)**

We can use two points to calculate the slope. Let's use the first two points (1, 25) and (3, 17).

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Here, \((x_1, y_1) = (1, 25)\) and \((x_2, y_2) = (3, 17)\):

\[
m = \frac{17 - 25}{3 - 1} = \frac{-8}{2} = -4
\]

2. **Find the y-intercept (b)**

Using the slope-intercept form \( y = mx + b \), we can substitute one of the points into the equation to find \( b \). We'll use the point (1, 25):

\[
25 = -4(1) + b
\]
\[
25 = -4 + b
\]
\[
b = 25 + 4 = 29
\]

3. **Write the Linear Function**

Now we have the slope \( m = -4 \) and the y-intercept \( b = 29 \). Thus, the equation that models the situation is:

\[
y = -4x + 29
\]

So the function rule in slope-intercept form is:

\[
y = -4x + 29
\]