Question

left on the card, y, as a function of the number of cups of coffee she buys, x.
(1) find and describe the rate of change of
the function.
(2) the rate of change is $-3$. it represents
a decrease in the amount on the gift
card for every cup nevaeh purchases.
find and describe the initial value of
the function.
(3) the initial value is $square$.

number of cups of coffee, $x$ | amount of money left on card ($), $y$
2 | 19
5 | 10
7 | 4

Explanation:

Step1: Calculate rate of change

Use two points $(x_1,y_1)=(2,19)$ and $(x_2,y_2)=(5,10)$.
Rate of change $=\frac{y_2-y_1}{x_2-x_1}=\frac{10-19}{5-2}=\frac{-9}{3}=-3$

Step2: Find initial value (x=0)

Use slope-intercept form $y=mx+b$, $m=-3$. Substitute $(2,19)$:
$19 = -3(2) + b$
$19 = -6 + b$
$b=19+6=25$

Answer:

(i)

The rate of change is $-3$. It represents a decrease of $\$3$ in the amount on the gift card for every cup Nevaeh purchases.

(ii)

The initial value is $\boldsymbol{25}$ (this is the starting amount on the gift card when 0 cups of coffee are purchased).