c What is the meaning of the x-intercept in this context?
A
aMost Money: Sibling C Least Money: Sibling A
B
bBiggest Spender: Sibling C Lowest Spender: Sibling A
C
cFirst: Sibling C Last: Sibling A
Practice makes perfect
a The amount of money each sibling is given is equal to the y-intercept of the function that models their spending behavior. This is because when they receive their money nothing has been spent yet. The time is then x=0.
Sibling A
Let's consider the graph that illustrates how Sibling A spends the money.
From the graph, we see that the y-intercept occurs at (0, 80), which means this sibling received $ 80.
Sibling B
Let's view the function that tells how Sibling B spends money.
y=-22.5x+ 90
From the function, we see that the y-intercept occurs at (0, 90). This sibling received $ 90.
Sibling C
In the table we do not have an ordered pair where x=0. But since the sibling spends the money at a constant rate, we can use the rate of change to find the y-intercept. We will use the Slope Formula and the points ( 1, 100) and ( 2, 75) to calculate the rate of change.
Sibling C has a rate of change of - 25, which corresponds to spending $25 each week. After one week of spending the sibling had $100 left. This means that the sibling must have started with $ 125.
Summarize
Now we can summarize and determine who received what amount of money.
Sibling A:& $ 80 ( least)
Sibling B:& $ 90
Sibling C:& $ 125 ( most)
b Comparing the spending rates means comparing the slopes of the spending functions. Let's do it!
Sibling A
To find the slope for Sibling A we can use the points from the graph, ( 2, 50) and ( 4, 20), and substitute them into the Slope Formula.
A slope of - 15 means that Sibling A spends their money at a rate of $ 15 per week.
Sibling B
For Sibling B, we have a function where we can identify the slope by looking at the coefficient of x.
y= - 22.5x+90
The slope for Sibling B is - 22.5, indicating a spending rate of $ 22.50 per week.
Sibling C
In Part A we determined the rate of change for Sibling C to be - 25. Thus, the spending rate is $ 25 per week.
Summarize
Let's summarize what we have found.
Sibling A:& $ 15 a week ( least)
Sibling B:& $ 22.50 a week
Sibling C:& $ 25 a week ( most)
c We can determine when all of the money has been spent by finding the x-intercept.
Sibling A
Sibling A spends $ 15 a week and starts out with $ 80. This means we can model their spending behavior with a linear function with a slope of - 15 and a y-intercept of 80.
y=- 15x+80We find the x-intercept by substituting 0 for y and solving for x.
Sibling A runs out of money after about 5.3 weeks.
Sibling B
We know that Sibling B has $25 left at the end of 4 weeks. Since they spend $25 a week, the remaining money will last one more week. Thus, Sibling B runs out of money after 5 weeks.
Sibling C
We have a linear function that tells us how much money Sibling C has at a given time.
y=- 22.5x+90
We find the x-intercept by substituting 0 for y and solving for x.
