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 $(3, 4)$.1. 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.2. find and describe the initial value of the function.the initial value is 16the container starts with 16 cups of milk.3. write an equation to represent the cups of milk left in the container, $y$.

Explanation:

Step1: Calculate rate of change

The rate of change (slope) $m$ between points $(x_1,y_1)=(1,12)$ and $(x_2,y_2)=(3,4)$ is $\frac{y_2-y_1}{x_2-x_1}$.
$m=\frac{4-12}{3-1}=\frac{-8}{2}=-4$

Step2: Find initial value (y-intercept)

Use slope-intercept form $y=mx+b$, substitute $m=-4$, $x=1$, $y=12$.
$12 = -4(1) + b$
$12 = -4 + b$
$b=12+4=16$

Step3: Write linear function

Substitute $m=-4$ and $b=16$ into $y=mx+b$.
$y=-4x+16$

Answer:

$y=-4x+16$