Question

Write a function rule, in slope-intercept form, that models this situation:
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 dispensed (x)
1
3
5
7

amount of pet food remaining (y)
25
17
9
1

Answer 1

To find the function rule in slope-intercept form (\(y = mx + b\)), we first need to determine the slope (\(m\)) and the y-intercept (\(b\)) using the data provided:

The meals dispensed (x) and the corresponding amount of pet food remaining (y) are as follows:

\[
\begin{align*}
1 & : 25 \\
3 & : 17 \\
5 & : 9 \\
7 & : 1 \\
\end{align*}
\]

### Step 1: Calculate the Slope (m)

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

The slope \(m\) is given by the formula:

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

Using the points (1, 25) and (3, 17):

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

### Step 2: Find the Y-Intercept (b)

We now use one of the points along with the slope to find the y-intercept \(b\). We'll use the point (1, 25).

The equation in slope-intercept form is:

\[
y = mx + b
\]

Substituting \(m = -4\) and the coordinates of the point (1, 25):

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

Solving for \(b\):

\[
25 = -4 + b \\
b = 25 + 4 \\
b = 29
\]

### Step 3: Write the Function Rule

Now that we have both \(m\) and \(b\), we can write the equation:

\[
y = -4x + 29
\]

### Conclusion

The function rule in slope-intercept form that models the situation is:

\[
y = -4x + 29
\]