Math 111 Chapters 1-3 Test
Retake
Fall14/CHutt
Name _________KEY__________
Directions: Open book, open notes and you may use your calculator.
Show your work to obtain partial credit if you miss a problem.
1. The following graph gives the annual median interest rate on passbook savings
accounts in the UK for the given years.
UK Bank Savings Rates
Interest Rate as a Percent
14
(1990, 13.56)
(1980, 10.5)
12
10
8
(2000, 5)
(1970, 5)
6
4
2
(2010, 2.8)
(1960, 3.38)
0
1950
1960
1970
1980
1990
2000
2010
2020
Year
Data: http://swanlowpark.co.uk/savingsinterestannual.jsp
a. For what decade is the average rate of change the highest?
What was the average rate of change for this decade? Looking at the
graph, I can sketch in lines to see which decade has the highest average rate of
change. I did that. From 1970 to 1980 I see the highest rate of change so:
10.5−5
1980−1970
=
5.5
10
= 𝟎. 𝟓𝟓
𝒑𝒆𝒓𝒄𝒆𝒏𝒕
𝒚𝒆𝒂𝒓
between 1970 and 1980.
b. For what decade did the average rate of change drop the most?
What was the average rate of change for this decade? Again I find where
the greatest rate of change occurred on the graph and use the average rate of
change formula (find the slope of the line for the two endpoints):
5−13.56
2000−1990
=
−8.56
10
= −𝟖. 𝟓𝟔
𝒑𝒆𝒓𝒄𝒆𝒏𝒕
𝒚𝒆𝒂𝒓
between 1990 and 2000.
2. Using the same graph as the first problem.
a. Find the best fit equation from 1960 to 1990 using linear regression on
your calculator.
1960 3.38
Show what you entered in your lists:
1970 5
1980 10.5
1990 13.56
Write your regression equation:
𝒚 = 𝟎. 𝟑𝟔𝟎𝟒𝒙 − 𝟕𝟎𝟑. 𝟔𝟖
What does the slope tell you about this data? (Hint:use the graph labels)
Interest increased at about 0.36% per year between 1960 and 1990.
What does the y-intercept tell you about this data? (Same hint)
That in year 0 interest rates were -703.68% which is really nonsensical.
This is extrapolation, and much has changed since year 0. 
b. Find the best fit equation from 1990 to 2010. Show your work, or show
the points you enter into your calculator.
1990 13.56
Show what you entered in your lists:
2000 5
2010 2.8
Write your regression equation: 𝒚 = −𝟎. 𝟓𝟑𝟖𝒙 + 𝟏𝟎𝟖𝟑. 𝟐
What does the slope tell you about this data? (Hint: use the graph labels)
Interest decreased at about 0.538% per year between 1990 and 2010.
What does the y-intercept tell you about this data? (Same hint)
In year 0 interest rates were 1083.2% which is again nonsense.
Extrapolation beyond where the data matters.
3. Some 10% alcohol solution is to be mixed with some 30% alcohol solution to
make 20 liters of 16% solution. How much of each must be used?
Step 1: Define variables for all your missing amounts:
Let 𝑥 be the amount of the 10% alcohol solution and 𝑦 the amount of the 30%.
Step 2: For mixture problems with two unknown amounts you must find two
equations.
Equation 1: The total volume equation: 𝑥 + 𝑦 = 20
Equation 2: The alcohol equation:
0.10𝑥 + 0.30𝑦 = 0.16(20) = 3.2
Step 3: Use substitution or elimination to solve.
Substitution: Solve for one variable in one equation:
𝑥 = 20 − 𝑦
Substitute into the second equation: 0.10(20 − 𝑦) + 0.30𝑦 = 3.2
Solve this equation: 2 − 0.10𝑦 + 0.30𝑦 = 3.2, 0.20𝑦 = 1.2, 𝑦 = 6
Thus 𝑥 = 14Elimination: Multiply one equation by a constant so you
can add equations:
Multiply : −10(0.10𝑥 + 0.30𝑦) = (3.2)(-10), −𝑥 − 3𝑦 = −32
Add the equations to eliminate a variable: −𝑥 − 3𝑦 = −32
𝑥 + 𝑦 = 20
−2𝑦 = −12
so 𝑦 = 6, and by substitution x=14.
Step 4: Check: 0.10(14) + 0.30(6) = 1.4 + 1.8 = 3.2 
Step 5: Answer the question.
14 liters of the 10% solution and 6 liters of the 30% solution were used.
1
4. Use transformations to sketch the graph of 𝑓(𝑥) = 4 − (𝑥 + 1)2 below
a. Write the domain in interval
notation. [−𝟏, ∞)
b. Write the range in interval notation.
(−∞, 𝟒]
c. Explain how each parameter
transforms the graph.
i. 4 ____is added after the square root, so it is a vertical shift of 4.
1
ii. Subtracting (𝑥 + 1)2 . __This is a horizontal reflection, so it turns the graph
“upside down.” ___
iii. Adding 1 to 𝑥 . __This is a horizontal shift to the left of 1.__________
5. Use the graph below to answer questions a - d.
Private versus Public School Tuition Growth
(2014, 37%)
Percentage change since 2004
40%
35%
30%
(2014, 24%)
25%
20%
15%
10%
5%
0%
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
Years
Private Colleges & Universities
Public Colleges & Universities
a. What is the average rate of change per year for public colleges &
universities from 2004 to 2014 for this graph?
37%−0%
2014−2004
= 𝟑. 𝟕%
b. Specifically, what does this average rate of change tell you about public
colleges & universities?
Tuition and fees at public colleges and university has gone up
about 3.7% per year for the last ten years.
c. What is the average rate of change per year for private colleges &
universities from 2004 to 2014 for this graph?
24%−0%
2014−2004
= 𝟐. 𝟒%
d. Specifically, what does this average rate of change tell you about private
colleges & universities?
Tuition and fees at private colleges and university has gone up
about 2.4% per year for the last ten years.
More
on next
page.
Use the graph below for e – h.
e. What is the average rate of change per year for public colleges &
universities from 2004 to 2014 for this graph?
18,391−13,376
2014−2004
= $𝟓𝟎𝟏. 𝟓𝟎
f. Specifically, what does this average rate of change tell you about public
colleges & universities?
The average rate of change of tuition and fees has gone up an average of
$501.50 per year.
g. What is the average rate of change per year for private colleges &
universities from 2004 to 2014 for this graph?
40,917−33,098
2014−2004
= $𝟕𝟖𝟏. 𝟗𝟎
h. Specifically, what does this average rate of change tell you about private
colleges & universities? The average rate of change of tuition and fees has
gone up an average of $781.90 per year.
EXTRA
on next
page.
Extra: For problem 5 carefully explain why public colleges and universities have a
higher rate of change than private colleges and universities on the first graph
while they have a lower rate of change than private colleges and universities on
the second graph.
While private school costs have gone up at a higher dollar rate on average over
the last 10 years, public schools have increased their tuition and fees at a greater
percentage of their original costs in 2004.