I only solved Option A in this answer, I think you needed to do both. You can do Option B in the same way!
2.
(a) You're supposed to find the pattern here. Use the slope-intercept form of the equation of a line. y=mx+b. In this situation, x is the week, and y is the amount paid. If you wanted, you could use whichever variables you wanted, of course. Basically, you need to find the slope, represented by m, which is the amount the amount paid increases by each week, and the y-intercept, b, which is the "amount paid" at week 0. (The y-intercept doesn't really make sense in this situation, but we'll get to that later.) You can use the formula m=(y2-y1)/(x2-x1). (x1,y1) and (x2,y2) are the input/output pairs. The x-coordinate is the week, and the y coordinate is the amount paid. Because the weeks are incrementing by 1, x2-x1=1. Because of this, you can just subtract the values and find out the slope is 50/week. After that, to complete the rule, you need to find the y-intercept. If the amount paid is 200 on week 1, and the amount paid increases by 50 each week, then you should be able to work backwards. 200-50=150
Another way to think about it is by plugging your points into the incomplete rule. Let b represent the y-intercept. Plug in the values for week 1 and solve for b.
Your work would probably look something like:
m=(y2-y1)/(x2-x1)
(x1,y1)=(1,200)
(x2,y2)=(2,250)
m=(250-200)/(2-1)
=50/1
=50
y=50x+b
200=50(1)+b
200=50+b
b=200-50
=150
y=50x+150
(b) If I understand the question correctly, you have to show two instances that prove that your rule work. Just pick any two weeks, like weeks 3 and 4 and show how the rule is correct.
y=50x+150
x=3
y=50(3)+150
=150+150
=300
(This is the same value that's in the chart, which proves that the rule works. Do the same thing with another week.)
3.
(a) The rules are functions because each input has only one output. You can test this out, it's true.
(b) So we did have to find the "y-intercept" earlier. But you have to look at the domain. There won't be any amount paid on Week 0, payment starts at Week 1. So you would have to respond that yes, they can be graphed, but there's no y-intercept; the graph starts at week 1. Aditionally, there are only data points at the integer values of x. Look at part c.
(c) The outputs are meaningless because you're counting the time by weeks. There won't be any amount paid in the middle of the week, so she can only use integer inputs.
I hope this helps!! This is my first time answering a question on here.
