How do you find the initial value of a linear function? What does it mean?
C
Find the greatest rate of change among the different proposals.
D
Whose proposal has the greatest rate of change?
A
Yes. See solution.
B
Yes. See solution.
C
Sarah's proposal. See solution.
D
Sarah's proposal. See solution.
Practice makes perfect
We are given Sarah's, Gene's, and Paul's proposals for the class fundraiser and asked to determine if these are represented by linear functions. Let's take a look at Sarah's proposal.
Since this graph is a straight line, it is a linear function. We can inspect Gene's proposal next.
In order to determine if this table represents a linear function, we need to compare the rates of increase for both the hours worked and the money raised.
We can see that the hours worked increase by 5 and that the money raised increases by 35. This means that Gene's proposal has a constant rate of change. Therefore, it is a linear function. Finally, we can consider Paul's proposal.
y=10x+7
Notice that this is a linear equation. Therefore, Paul's proposal is a linear function as well. All three proposals are linear functions!
We can tell that the class has an initial amount of money by looking at their proposals. Paul's proposal is given by a linear equation.
y=10x+7
If we let x be equal to 0, we will find out how much money has been raised after 0 hours of working — that is, we will find the initial value before any work is done.
This means that the class initially has $7 in their account. Please note that we could have also reached the same conclusion by inspecting Gene's or Sarah's proposals.
We will now find the rate of change of each proposal. Let's consider each proposal, one at a time!
Sarah's Proposal
To find Sarah's proposal's constant rate of change, we use two points of the graph.
Next, we find the ratio of the rise to the run, or slope, of the line.
m=80-20/6-1
Let's do it!
The slope is 12. This mean that Sarah's proposal has a constant rate of change of 12.
Gene's Proposal
Now we will find the rate of change of Gene's proposal. We found in Part A that for every 5 hours worked, the money raised increases by 35.
The constant rate of change can be found by dividing 35 by 5 and simplifying.
m=35/5 ⇓
m=7
Therefore, Gene's proposal has a constant rate of change of 7.
Paul's Proposal
Finally, we will find the constant rate of change of Paul's proposal. This can be done by looking at the coefficient of the x-variable in the equation.
y= 10x+7
The coefficient of x is 10, which means that the constant rate of change of Paul's proposal is 10.
Summary
We will now summarize our findings.
Constant Rate of Change
Sarah's Proposal
Gene's Proposal
Paul's Proposal
12
7
10
Sarah's proposal has the greatest constant rate of change. This means that it raises money at the fastest rate.
Sarah and her classmates made these proposals because they are hoping to raise $200. Let's find out how much time each proposal will take to raise this amount!
Sarah's Proposal
We can extend the graph of Sarah's proposal in order to see when the money raised will be equal to 200. We begin by extending the coordinate plane.
Next, we extend the linear function with the help of a ruler. This will let us find at which x-value the y-value is equal to 200.
The graph of Sarah's Proposal passes through ( 16, 200). This means that the class will raise $ 200 after working 16 hours if they choose this proposal.
Gene's Proposal
We can also extend Gene's proposal if we keep adding rows to its table.
If the class follows Gene's proposal they will need between 25 and 30 hours to reach their goal!
Paul's Proposal
We can find how many hours it will take the class to reach $200 using Paul's proposal by setting y equal to 200 and solving the resulting equation for x. Let's do it!
This means that the class will take 19.3 hours to meet their goal following Paul's Proposal.
Summary
Of all three proposals, Sarah's proposal reaches the goal in the least amount of time, which is 16 hours. This makes sense, as we also found in Part C that Sarah's proposal raises money at the fastest rate, meaning fewer hours must be worked to raise the same amount of money. Therefore, they should follow Sarah's proposal.
