Question

santiago buys a container of milk to make atole. the cups of milk left in the container, y, is a linear function of the number of batches he makes, x. the graph of the function passes through the points (1,12) and (3,4). find and describe the rate of change of the function. the rate of change is -4. it represents a decrease in the amount of milk in the container for each batch of atole made. find and describe the initial value of the function. the initial value is 16. the container starts with 16 cups of milk.

Explanation:

Step1: Find the rate - of - change (slope)

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Given the points $(x_1,y_1)=(1,12)$ and $(x_2,y_2)=(3,4)$. Then $m=\frac{4 - 12}{3 - 1}=\frac{-8}{2}=-4$. It represents a decrease in the amount of milk in the container for each batch of atole made.

Step2: Find the initial value (y - intercept)

The equation of a line is $y=mx + b$. We know $m=-4$, and we can use the point $(1,12)$. Substitute into the equation: $12=-4\times1 + b$. Solve for $b$: $b=12 + 4=16$. The initial value is 16, which means the container starts with 16 cups of milk.

Answer:

The rate of change is - 4. It represents a decrease in the amount of milk in the container for each batch of atole made.
The initial value is 16. The container starts with 16 cups of milk.