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:
\[ y = -4x + 29 \]