Question

convenience store
your friends know that you stop at the local convenience store every morning on your way to school and get a bottle of juice. for your birthday, they get you a $50 gift card. suppose that a small bottle of juice costs $2 and all you ever buy at the store is juice.

1. (a) write a function, f, that can be used to track your gift card balance in terms of the number of bottles of juice, x, you have purchased.

(b) what is the rate of change of this function? what does this mean for this scenario?
(c) what will the graph of the function look like? how do you know?
(d) what does “f(2)=46” mean in the context of the problem?
(e) using the format given below, complete the table representing your function by choosing several different values representing the number of bottles of juice you have purchased and computing the remaining balance on your gift card. then, use your table to construct ordered pairs of the form (x, f(x)).

Explanation:

Step1: Find the function for gift - card balance

The initial balance is $50$ and each bottle of juice costs $2$. So the balance $f(x)$ after buying $x$ bottles is $f(x)=50 - 2x$.

Step2: Determine the rate of change

The function $f(x)=50 - 2x$ is in the form $y = mx + b$ where $m$ is the slope. Here $m=- 2$. The rate of change is - 2. It means that for each bottle of juice purchased, the gift - card balance decreases by $2$.

Step3: Describe the graph

The function $f(x)=50 - 2x$ is a linear function. Since the slope $m=-2<0$ and the $y$ - intercept $b = 50$, the graph is a straight line that slopes downwards from left to right, starting at the point $(0,50)$ on the $y$ - axis.

Step4: Interpret $f(2)=46$

When $x = 2$ (2 bottles of juice are purchased), $f(2)=50-2\times2=46$. It means that after buying 2 bottles of juice, the remaining balance on the gift card is $46$.

Step5: Complete the table

$x$ (number of bottles)	$f(x)=50 - 2x$ (balance)	Ordered pair $(x,f(x))$
1	$f(1)=50-2\times1 = 48$	$(1,48)$
2	$f(2)=50-2\times2 = 46$	$(2,46)$
3	$f(3)=50-2\times3 = 44$	$(3,44)$
4	$f(4)=50-2\times4 = 42$	$(4,42)$
5	$f(5)=50-2\times5 = 40$	$(5,40)$

Answer:

(a) $f(x)=50 - 2x$
(b) Rate of change: - 2. It means the gift - card balance decreases by $2$ for each bottle of juice purchased.
(c) A straight - line sloping downwards from left to right, starting at $(0,50)$ on the $y$ - axis, because it is a linear function with a negative slope and a $y$ - intercept of 50.
(d) After buying 2 bottles of juice, the remaining balance on the gift card is $46$.
(e) See the table above for values of $x$, $f(x)$ and ordered pairs.