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:

Answer 1

To write a function in slope-intercept form (\(y = mx + b\)), we need to identify the slope (\(m\)) and the y-intercept (\(b\)).

Since I don't have the actual data from the table you mentioned, let's outline the steps you would typically follow to derive this rule:

1. **Identify Points from the Table**: Look for at least two data points in the table that represent the number of meals dispensed (\(x\)) and the amount of pet food remaining (\(y\)) in cups.

2. **Calculate the Slope (\(m\))**: Use the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \((x_1, y_1)\) and \((x_2, y_2)\) are two points from the table.

3. **Find the y-Intercept (\(b\))**: The y-intercept is the value of \(y\) when \(x = 0\). If the table includes a point where \(x = 0\), you can use that directly. Otherwise, you can use the slope and one of the points to find it using:
\[
b = y - mx
\]

4. **Write the Function**: Once you have \(m\) and \(b\), plug these values into the slope-intercept form:
\[
y = mx + b
\]

### Example
Assuming from the table you found these points:
- Point 1: (1, 8) → After 1 meal, 8 cups remain.
- Point 2: (3, 4) → After 3 meals, 4 cups remain.

Calculate the slope (\(m\)):
\[
m = \frac{4 - 8}{3 - 1} = \frac{-4}{2} = -2
\]

Next, use one of the points to find the intercept (\(b\)):
Using point (1, 8):
\[
8 = -2(1) + b \implies 8 = -2 + b \implies b = 10
\]

Now you can write the function:
\[
y = -2x + 10
\]

This formula indicates that for each meal dispensed, 2 cups of food are used, starting from 10 cups.

If you provide the specific data from your table, I can help you derive the function rule directly!