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The Ivy 97 Exponential Functions

9. December 2017 11. October 2019
c(t) = 1,870(1.025)t The temperature of a cup of cocoa t minutes after it is made
is given by the function C(t) = a + b(0.95)t, where a is the
The function above models the number of capybaras, c(t), temperature of the room, in degrees Celsius, and a + b is the
living in a section of tropical forest in Brazil, where t is the initial temperature of the cocoa, in degrees Celsius. If the
number of years since 2010. Which of the following is the cocoa is initially 80 °C and the temperature of the room is 20
best interpretation of the value 1.025 in the context? °C, what will the temperature of the room is 20 °C, what will
the temperature of the cocoa be 2 minutes later, rounded to
a) Each year, the number of capybaras is increasing by the nearest degree Celsius?
approximately 1 from the preceding year.
a) 72 °C
b) Each year, the number of capybaras is increasing by b) 74 °C
approximately 25 from the preceding year. c) 78 °C
d) 90 °C
c) Each year, the number of capybaras is increasing by
approximately 2.5% from the preceding year.

d) Each year, the number of capybaras is increasing by
approximately 125% from the preceding year.

10. 12. October 2019 US
Poultry should be cooked to a temperature of 75 C . A
chicken is removed from the oven and left to rest in a room Kao measured the temperature of a cup of hot chocolate
that is at a constant temperature of 22 C . The temperature of
the chicken t hours after it is removed from the oven is given placed in a room with a constant temperature of 70 degrees
by the exponential function:
Fahrenheit (°F). The temperature of the hot chocolate was
T(t) = 22 + 53(0.74)t
What is the approximate temperature of the chicken after 2 185 °F at 6:00 p.m. when it started cooling. The temperature
hours?
of the hot chocolate was 156 °F at 6:05 p.m. and 135 °F at
a) 22 °C
b) 51 °C : p.m. The hot chocolate’s temperature continued to
c) 74 °C
d) 75 °C decrease. Of the following functions, which best models the

temperature T(m), in degrees Fahrenheit, of Kao’s hot

chocolate m minutes after it started cooling?

a) T(m) = 185(1.25)m

b) T(m) = 185(0.85)m

c) T(m) = (185 – 70)(0.75) m
5

d) T(m) = 70 + 115(0.75) m
5

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Exponential Functions 98 The Ivy

13. October 2019 16.

The amount of fermium-253 in a sample is reduced by half After 24 hours, 30%of a 25-milligram dose of a new antibiotic

every 3 days because of radioactive decay. If there is an remains in the body. Which of the following functions, M,

original mass M0, in milligrams, of fermium-253 in the models the amount of the antibiotic ( in milligrams) that
sample, which of the following equations gives the mass, M,
remains in the body after h hours?

in milligrams of fermium-253 that remains after t days? a) M(h) = 30 (0.3) h
24

a) M = M0 ( 1 ) 2t b) M(h) = 25 (0.3) h
3 24

b) M = M0 ( 1 ) t c) M(h) = 25(0.3)h
3 2

c) M = M0 ( 1 )3t d) M(h) = 25 (0.7) h
2 24

d) M = M0 ( 1 ) t
2 3

17.

14. November 2017 The following function, L, gives the approximate percent
Starting in 1970, the population of a city doubled every 10
years until 2010.The population of the city was 36,000 in literacy rate in India t years after 1900.
1970. Which of the following expressions gives the
population of the city in 2010? L(t) = 5.31.025t

a) 36,000(2)4 Which of the following equivalent functions shows, as a
b) 36,000(2)10
c) 36,000(2)40 constant or coefficient, the approximate number of years it
d) 36,000(2)(40)
took for the literacy rate to triple?
15. May 2019
S(n) = 38,000an a) L(t) = 5.3 3 t
The function S above models the annual salary, in dollars, of 44.5
an employee n years after starting a job, where a is a constant.
If the employee’s salary increases by % each year, what is b) L(t) = 5.3 1.077 t
the value of a? 3
a) 0.04
b) 0.4 c) L(t) = 5.3·1.0083t
c) 1.04
d) 1.4 d) L(t) = 3·1.025t + 23

18. March 2019
A microbiologist is growing mammalian cells in a liquid
culture. At the beginning of each day, there are twice as many
cells in the culture as there were at the beginning of the
preceding day. If there are 100 cells at the beginning of day l,
how many cells will be in the culture at the beginning of day
6?

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The Ivy 99 Exponential Functions

19. 21. June 2016
Shriya began growing a colony of bacteria in a culture. The A radioactive substance decays by approximately the same
following function, P, gives the population in millions of percentage each year. A function of which of the following
bacteria in the culture after growing for t minutes. forms best models the amount, in milligrams, y, of the
substance remaining x years from now, where m and p are
P(t) = 3 100.015t positive constants?

Approximately how long will it take for Shriya's bacteria a) y = px
colony to grow to 100 times its original population? Round b) y = px – m
to the nearest minute. c) y = (1 + p)x
d) y = m(1 – p)x
a) 133 minutes
b) 67 minutes
c) 33 minutes
d) 10 minutes

22. March 2018

Before its launch, the first stage of the Saturn V rocket held

about 540,000 gallons of fuel. The number of gallons of fuel

20. March 2019 F(t) remaining t minutes after its launch can be modeled by
19
T(x) = 24(0.97)x the function F(t) = 540,000 ( 21 ) 20t, where t ≤ 6. Which of

A sample of bacteria in a liquid growth medium that has an the following describes the meaning of the fraction 19 in
21
initial temperature of 24 degrees Celsius (°C) is placed in a

refrigerator with the temperature set to 0 °C. The function the context described?

above approximates the temperature T, in °C, of the medium a) The fraction of the 540,000 gallons of fuel the

x minutes after being placed in the refrigerator. Which of the rocket had at its launch

following best approximates the amount that the temperature, b) The fraction of the 540,000 gallons of fuel the

in °C, of the medium has decreased after 20 minutes? rocket had 3 seconds after its launch

a) T(0) c) The fraction of the 540,000 gallons of fuel the

b) T(20) rocket had 20 seconds after its launch

c) T (20) d) The fraction of the 540,000 gallons of fuel the
T (0)
rocket had 1,200 seconds after its launch
d) T(0) – T(20)

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Exponential Functions 100 The Ivy

23. March 2018 25. Level 3 (Manipulating Exponential Expressions)

The volume of water in a reservoir at the start of a The amount of money, A, in dollars that is in Antoine's bank

construction project is V0. For every 10-day period, the account after t years can be shown with the function below.
volume of water in the reservoir will decrease by 3% of the
A(t) = 25001.005t

volume from 10 days before. Which of the following Which function represents the amount of money in Antoine's

equations represents the amount of water left in the bank account after m months?
0.005
reservoir, Vt, after t days? a) A(m) = 2500∙ (1 + 12 )m

a) Vt = V0(0.97)t b) A(m) = 2500∙1.0051m2

b) Vt = V0(0.97) t 1.005
10 12

c) Vt = V0(0.03)10t c) A(m) = 2500∙ ( )m

d) Vt = t (0.97 V0) d) A(m) = 2500 ∙1.005m
10 12

24. 26. May 2018

P(t) = 60(3) t P = 215(1.005) t
2 3

The number of microscopic organisms in a petri dish grows The equation above can be used to model the population, in

exponentially with time. The function P above models the thousands, of a certain city t years after 2000. According to

number of organisms after growing for t days in the petri dish. the model, the population is predicted to increase by 0.5%

Based on the function, which of the following statements is every n month. What is the value of n?

true? a) 3

a) The predicted number of organisms in the dish b) 4

triples every two days. c) 12

b) The predicted number of organisms in the dish d) 36

doubles every three days.

c) The predicted number of organisms in the dish

triples every day.

d) The predicted number of organisms in the dish

doubles every day.

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The Ivy 101 Exponential Functions

27. 29.
Abhishek received an injection of a medication. The
The Golden Years Senior Citizen Center uses a phone tree to following expression shows the number of milligrams of
medication left in his body after t hours.
announce when the center will be closed for poor weather.
37 0.251t6
When each person receives a phone call, that person has a list
Half-life of a medicine refers to the amount of time it takes
of three more people to call. The function c approximates the for half of the medicine to leave the body. What is the half-
life (in hours) of Abhishek's medicine?
total number of calls made after m minutes since the start of

the phone tree.

c(m) = 3 (31m0 −1)
2
Approximately how many minutes will it take for the number

of calls to reach 363?

28. 30.
Lúcia posted a video of her cat playing on the piano. She
An ultrasound technician sometimes uses microbubbles as a found that the following expression modeled the total number
of people who had viewed her video t days after she posted it.
contrast agent to improve the clarity of ultrasounds. However,
18  160.0125t
the technician has a limited time to work, since the ultrasound
After how many days did the total number of people viewing
process quickly destroys the microbubbles. the video double from the original number of people? Round
to the nearest day.
The following expression represents the concentration, in

microbubbles per milliliter, of the microbubble suspension

remaining in a client t seconds after the technician has begun

the ultrasound process.

100,000,000 0.512t0

Which expression shows the multiple of the concentration

remaining each minute as a constant or coefficient?

a) 100,000,000  0.5 t
60

b) 100,000,000∙ 0.5 t
60

c) 100,000,000∙0.560t

d) 100,000,000∙(0.52)60t

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Exponential Functions 102 The Ivy

31. Level 3 (Quadratic and Exponetial Word Problems)

Evolutionary medicine publications each year The number of medical publications that mention evolutionary

Number of publications Year medicine each year is shown at left, with the data for 2010 and
21 2000 2011 missing. Based on the entire trend of the data, what is the best
29 2001 estimate of the number of publications that mentioned evolutionary
40 2002 medicine in 2011?
57 2003
a) 120
b) 410

79 2004 c) 670

111 2005 d) 840

156 2006

218 2007

305 2008

427 2009

32. Level 3 (Quadratic and Exponetial Word Problems)

c At a party, there is a cake of length c, as shown at left. Each person at the party

s = 0 takes a slice of cake which is exactly half the length of whatever cake is remaining.
a(s) represents the total length of cake taken after s people have taken a slice. Which
a(0) = 0
s=1 of the following equations represents a(s)?

a(1) ( )a) 1 s
s=2 a(s) = c∙ 1− 2 

a(2) ( )b) 1 2
s=3 a(s) = c∙ 1− s 

a(3) ( )c) 1s
a(s) = c∙ 2

( )d) 12
a(s) = c∙ s

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The Ivy 103 Dividing Polynomial, Function

Dividing Polynomials by a Linear Expression

1. 2.
Divide 2x3 – 3x2 – 3x + 7 by x – 2. Divide 2x3 – 7x2 – 16x + 11 by x – 5.

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Dividing Polynomial, Function 104 The Ivy

3. 6.

When 6x2 – 5x + 4 is divided by 3x + 2, the result is When 5x + 3 is divided by x + m, where m is a constant, the

2x – 3 + 3 R 2 , where R is a constant. What is the value of result can be written as 5 + r m . What is r in terms of m?
x+ x+

R? a) 3 – 5m

b) 3m – 5

c) 5m + 3

d) 8m

7.

When 3x2 + x + 2 is divided by x – 1, the result can be

4. expressed as ax + b + x c 1 , where a, b, and c are constants.
The function f is defined by a polynomial. If f (0) = 3, which
of the following must be true about f (x)? −

a) The remainder when f (x) is divided by x + 3 is 0. What is the value of a + b + c?
b) The remainder when f (x) is divided by x – 3 is 0.
c) The remainder when f (x) is divided by x is –3. 8.
d) The remainder when f (x) is divided by x is 3. y

5. – – 10 20 x
f (x) = ax3 + bx2 + cx + d
In the polynomial above, a, b, c, and d are constants. For a function g, the graph of y = g(x) is shown at left. When
If f (–5) = 3, which of the following must be true about f (x)? g(x) is divided by (x + 10), the remainder is –20. Which of
a) x – 3 is a factor of f (x). the following is closest to the remainder when g(x) is
b) The remainder when f (x) is divided by x + 5 is 3. divided by (x – 10)?
c) x + 2 is a factor of f (x).
d) x + 5 is a factor of f (x). a) –28
b) –2
c) 2
d) 28

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The Ivy 105 Dividing Polynomial, Function

9. March 2019 US 12.

f (x) = (x + 4)(x – 1)(2x – 3) x3 + 25x2 + 50x – 1000
The polynomial above has (x – 5) and (x + 10) as factors.
The function f is defined above. Which of the following is What is the remaining factor?

NOT an x-intercept of the graph of the function in the xy- a) (x + 2)
b) (x – 2)
plane? c) (x + 20)
d) (x – 20)
a) (–4, 0)
2
b) ( − 3 , 0)

c) (1, 0)
3
d) ( 2 , 0)

10. March 2019 US 13. May 2019 US

h(x) = x3 + ax2 + bx + c 2x3 + 11x2 + 5x
The function h is defined above, where a, b, and c are Which of the following is NOT a factor of the polynomial
integer constants. If the zeros of the function are –5, 6, and above?
7, what is the value of c?
a) x
b) x + 5
c) 2x + 1
d) 2x + 5

11. October 2018 14. March 2018 US

In the xy-plane, the graph of the cubic equation P(x) = x2 – 11x + k
y = x3 + ax2 + bx + c, where a, b, and c are constants, has x- In the function above, k is a constant. If 2 is a zero of the
intercept at x = –4, x = –5, and x = –7. What is the value of function, what is the value of k?
a?
a) –18
b) –2
c) 3
d) 18

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Dividing Polynomial, Function 106 The Ivy

15. April 2017 18. June 2016

The polynomial p4 + 4p3 + 3p2 – 4p – 4 can be written as The points (–3, 0), (1, 0), and (2, 0) all lie in the xy-plane on
(p2 – 1)(p + 2)2. What are all of the roots of the polynomial? the graph of the polynomial function f. Which of the
following could define f ?
a) –2 and 1
b) –2, 1, and 4 a) f (x) = (x + 1)(x + 2)(x – 3)
c) –2, –1, and 1 b) f (x) = (x + 1)(x – 2)(x + 3)
d) –1, 1, and 2 c) f (x) = (x – 1)(x + 2)(x + 3)
d) f (x) = (x – 1)(x – 2)(x + 3)

16. March 2018 19. December 2018

In the xy-plane, the four x-intercept of the graph of the The graph of a polynomial equation in the xy-plane contains
polynomial function f are –2, 3, 0, and 1. Which of the the points (–1, 0), (2, 0), and (3, 0). Which of the following
following is a factor of f ? could be the equation of the graph?

a) x2 – x a) y = –x(2x)(3x)
b) x2 – x – 2 b) y = (x – 1)(x + 2)(x + 3)
c) x2 – x + 6 c) y = x(x – 1)(x – 2)(x + 3)
d) x2 – 4x – 3 d) y = (x + 1)2(x – 2)(x – 3)

17. May 2018 US 20. October 2016

y = (x – h)2(x + h)(x + k) f (x) = (x – 3)(x2 – 4x + 3)
The equation above is graphed in the xy-plane. If h and k are How many distinct zeros does the function f, defined?

positive constants and h  k, how many distinct x-intercepts a) None
b) One
does the graph have? c) Two
a) 1 d) Three
b) 2
c) 3
d) 4

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The Ivy 107 Dividing Polynomial, Function

21. 24. October 2018

G(z) = (z – 1)5 – (z – 1)4 The function b is defined by b(x) = (x2 – 1)(x + 1).
The polynomial function G is defined above. What is the Which of the following represents the graph of y = b(x) in
product of the zeros of G? the xy-plane?

a) y
3

22. – 3x
– 3x
A third degree polynomial function y = g(x) is defined so
2 b) y
that g( 3 ) = 0 and g(0) = 5. Which of the following must be 3

a factor of g? –
–
a) x – 5
c) y
b) x + 5 3

c) 3x – 2

d) 3x + 2

23. May 2019 US – 3x
– 3x
x p(x)
–2 5 d) y
–1 0 3
0 –3
1 –1 –
20 –
The table above gives selected values of a polynomial
function p. Based on the values in the table, which of the
following must be a factor of p?
a) x – 3
b) x + 3
c) (x – 1)(x + 2)
d) (x + 1)(x – 2)

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Dividing Polynomial, Function 108 The Ivy

25. May 2018 27.
y
y
30 15

20 10

10 5
– – O 2 4x
123456 x
The graph of the function f is shown in the xy-plane above, ––––
where y = f (x). Which of the following could define f ?
–
a) f (x) = (x – 3)(x – 1)2(x + 2)2
b) f (x) = (x – 3)2(x – 1)(x + 2) –
c) f (x) = (x + 3)(x + 1)2(x – 2)2
d) f (x) = (x + 3)2(x + 1)(x – 2) If a is a positive constant, which of the following equations
could represent the polynomial graphed in the xy-plane at
26. left?
y
a) y = a(x – 3)(x + 4)2
b) y = a(x + 3) (x – 4)2
c) y = a(x – 3)2(x + 4)
d) y = a(x + 3)2(x – 4)

x 28. April 2019 US

Which of the following functions could represent the graph y
at left in the xy-plane, where y = P(x)?
20
a) P(x) = x2 + 6x + 8 15 y = f (x)
b) P(x) = x3 + 6x2 + 8x 10
c) P(x) = x2 – 6x + 8 5
d) P(x) = x3 – 6x2 + 8x O 12 3 4 5678 9 x
–

The graph of the cubic function f is shown in the xy-plane
above. If f (a) = 0, where a is a constant, what is one
possible value of a?

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The Ivy 109 Dividing Polynomial, Function

29. May 2018 31.

y Which of the following graphs could represent the equation
5 y = p(x)
y = –(x + 2)3?
4
a) y
3 8642

2 – – – 24 68
–––
1 x

– – – – – O 1 2 3 4 5x
–

–

The graph of y = p(x) is shown in the xy-plane above. Which b) y
of the following equations could define the function p? (May
2019) 8642 24 68 x
–––
a) p(x) = (x – 2)(x + 1)2
b) p(x) = (x – 2)2(x + 1) –––
c) p(x) = (x + 2)2(x – 1)
d) p(x) = (x + 2)(x – 1)2 c) y

30. 8642 x
–––
y 24 68
–––
7
654321 d) y

––– 1234567 x 8642 x
–––– –––
– 24 68
–––

Given some rational constant a, which polynomial equation
could represent the graph at left in the xy-plane?

a) y = a(x + 1)2(x – 5)
b) y = a(x + 1)(x – 5)
c) y = a(x + 1)(2x + 1)(x – 5)
d) y = a(x – 1)(x + 5)2

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Dividing Polynomial, Function 110 The Ivy

Function Notation

1. November 2017 4. October 2017 US
The function f is defined by f (x) = 3x2 – 8x + 17. If f (x) = 5x2 – 3 and f (x + a) = 5x2 + 30x + 42, what is the
What is the value of f (–2)? value of a?

2. a) –30
If f (x) = x2 – 3x + 5, then f (2x) = b) –3
c) 3
a) 4x2 – 6x + 5 d) 30
b) 2x2 – 6x + 10
c) 2x2 – 6x + 5 5. April 2019 US
d) 4x2 + 6x + 5 f (x) = 2(x – 1) + 2

3. For the function f defined above, what is the value of f (1)?
a) 3
Let the function g be defined by g(x) = x 3 − 7 . If g(k) = 57, b) 2
4 c) 0
d) –1
then k =
6. May 2018 US
The function f has the property that, for all x, 3f (x) = f (3x).
If f (6) = 12, what is the value of f (2)?

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The Ivy 111 Dividing Polynomial, Function

7. December 2017 10. December 2018
P(x) = x4 – 5x3 – 5x + 6 + 4k
The function p is defined by p(h) = 2h + h2.
In the polynomial above, k is a constant. If P(2) = –20, what If k > –2 and p(k) = p(–2), what is the value of k?
is the value of k?
a) –1
a) –2 b) 0
b) 1.5 c) 1
c) 2 d) 2
d) 7

8. 11.

If the function k is defined by k(h) = (h + 1)2, then k(x – 2) = Let the function f be defined by f (t) = 6t for all numbers t.
a) x2 – x
b) x2 – 2x + 1 Which of the following is equivalent to f (t – r)?
c) x2 + 2x – 1
d) x2 – 1 a) t−r
6

b) 6t – r

c) 6t – 6r

d) 12(t – r)

9. December 2018 12.
Two functions f and g are such that f (x) = 1 + 2x and
For the function f, if f (z) = 3z, which of the following is g(f (x)) = x + 3. What is the value of g(5)?
equivalent to f (z – 2)?

a) –3z
b) 3z – 2
c) 3z – 6
d) –3z + 6

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Dividing Polynomial, Function 112 The Ivy

13. 16. August 2020

If f (x) = x + 2 and f (g(1)) = 6, which of the following could The function f is graphed in the xy-plane, where
be g(x)?
y = f (x). The graph contains the points (0, 3), (6, 0), and
a) x + 3
b) x – 3 (9, 0). Which of the following CANNOT be true?
c) 2x + 1
d) 2x – 1 a) f (9) < 0
b) f (3) < 0
c) f (3) > 0
d) f (0) < 9

14. October 2018 17. November 2017

The function f is defined by f (x) = x. If a and b are x f (x) g(x)
constants, which of the following must be true? 140
4 6 12
I. f (a + b) = f (a) + f (b) 647
II. 2 f (a) = f (2a) 9 1 10
III. f (–a) = –f (a) 12 9 8
The table above shows several values of function f and
a) II only function g. According to the table, if f (4) = a, what is g(a)?
b) I and III only
c) II and III only
d) I, II, and III

18.

x f (x) g(x)

15. October 2018 –2 –6 –5

If the function p is defined by p(x) = x2 + 3x and k is a –1 2 –2
nonzero constant such that p(2k) = 8k, what is the value of
k? 23 4

7 7 11

Consider the table shown above. What is the value of

(g o f )(–1)?

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The Ivy 113 Dividing Polynomial, Function

x f (x) g(x) 21.
123
245 x f (x)
351 1a
432 2b
514 3c
For the function f, the table above shows some values of x
Example 19 and 20 refer to the accompanying table, which and their corresponding values of f (x) in terms of the
gives the values of functions f and g for integer values of x constants a, b, and c. If a < b < c, which of the following
from 1 to 5, inclusive could NOT be the graph of y = f (x) in the xy-plane?

19. a) y
According to the table, if f (5) = p, what is the value of
g (p)? 1x

a) 1 b) y
b) 2
c) 3
d) 4

1 x
1
20.
Function h is defined by h(x) = 2f (x) – 1, where function f is c) y
defined in the accompanying table. What is the value of g(k) d)
when h(k) = 5? 1 x
1
a) 1
b) 2
c) 3
d) 4

y

1 x
1

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Dividing Polynomial, Function 114 The Ivy

Function Graph

1. November 2017 3. May 2017 US
The entire graph of y = f (x) is shown in the xy-plane below.
y x g(x)
y –1 –2
y = f (x)
02
4 1 x 10
2 O1 2 –1
31
– – – –O 1234x
– The graph of the function f is shown in the xy-plane above,

and selected values for the function g are shown in the table.

Which of the following could be the product of all the values For which of the following values of x is g(x) > f (x)?

of x for which f (x) = 0? a) 0
b) 1
a) –2 c) 2
d) 3
b) 0

c) 3
2
2
d) 3

2. June 2016 4. October 2017 US
y

y

y = f (x) y = f (x)

1 x 1 x
O1 O1

In the xy-plane above, the graph of the function f on the The figure above shows the complete graph of the function f
in the xy-plane. The function g (not shown) is defined by
interval –7  x  6 is shown. For which of the following g(x) = f (x) + 6. What is the maximum value of the function
g?
values of c is f (c – 4) = c?
a) 0
b) 1
c) 3
d) 5

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The Ivy 115 Dividing Polynomial, Function

5. March 2019 6. May 2019

Stopping distance (m)Stopping Distance y
140 4
120
100 – 4x
80
60 –
40
20 The graph of y = f (x) is shown in the xy-plane above.

0 0 20 40 60 80 100 120 Which of the following could define f?
1
Vehicle speed (km/hr) a) f (x) = x−1 – 1
The graph above shows the quadratic function f.
The stopping distance f (x), in meters (m), is the distance b) f (x) = 1 +1
required to stop a vehicle based on the speed x, in x−1
kilometers per hour (km/hr), of the vehicle. Which of the 1
following is the best interpretation of the equation c) f (x) = x+1 –1
f (91) = 110 in the context of the problem?
d) f (x) = 1 +1
a) A vehicle traveling 91 km/hr requires 110 m to x+1
stop.
7. October 2018
b) A vehicle traveling 110 km/hr requires 91 m to y
stop.

c) The vertex of the graph of f is (91, 110).
d) The x– and y– intercepts of the graph of f are 91

and 110, respectively.

20
y = f (x)

10

Oa b c d x

The complete graph of the function f is shown in the xy-
plane above. What is the value of f (a) – f (c)?

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Dividing Polynomial, Function 116 The Ivy

8. September 2020 10.

y y
10
y = f (x) 8 9 y = f (x) y = g (x)
6
8

4 7

2 6
5

– – – –O 2 4 6 8x 4
– y = f (x) 3

– 2

1
0 1 2 3 4 5 6 7 8 9 10 x

– Let the function f and g be defined by the graph in the

– accompanying diagram. What is the value of f (g(3))?

The graph of y = f (x) is shown. If f (a) = 0, which of the 11.
following could be the value of a?

a) –1
b) –2
c) –3
d) –4

9. y = f (x) y
y y = g(x)
6
5 x
4
3
2
1

1 y = h(x) –4 –3 –2 –1–1 12 3 4
01 x
The graph of functions f and g for values of x between –3
The figure above shows the graph of y = h(x). If the function
x and 3 are shown above. Which of the following describes
k is defined by k(x) = h ( 2 ) + 3, what is the value of k(–2)?
the set of all x for which g(x)  f (x)?

a) x  –3
b) –3  x  –1 or 2  x  3
c) –1  x  2
d) 1  x  6

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The Ivy 117 Dividing Polynomial, Function

12. 13.

y y = f (x) y
5
9 4 y = f (x)
8 3
7 2
6 1
5 –5 –4 –3 –2 –1–1 1 2 3 4 5 x
4 –2
3 –3
2 –4
1 –5

– – – – – –– 1 2 3 4 5 6 x Let function f be defined by the accompanying graph.

– If f (–4) + f (2) = f (p), what is the value of p?
–
a) –2
The figure at left shows the graph of a quadratic function f b) 0
with a minimum point at (1, –2). If f (5) = f (c), then which of c) 1
the following could be the value of c? d) 1.5

a) –5
b) –3
c) 0
d) 6

14. y

y x
g(x) f (x)

x

The graphs of g and f are shown above. What is the value of f (g(3))?
a) 1 b) 3
c) 4 d) 5

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Dividing Polynomial, Function 118 The Ivy

Translation

1. December 2018 3.
An equation of a line in the xy-plane is y = mx + b, where m The graph of function f can be shifted 4 units to the left to
and b are constants. Which of the following changes to the obtain the graph of function g. If f (x) = x3, which of the
equation would correspond to a vertical shift of the line by 3 following is equivalent to function g?
units up?
a) g(x) = x3 – 4
a) Increasing the value of m by 3 b) g(x) = x3 + 4
b) Increasing the value of b by 3 c) g(x) = (x – 4)3
c) Increasing the values of both m and b by 3 d) g(x) = (x + 4)3
d) Increasing the values of both m and b by 1.5
4.
2. The graph of y = f (x) can be shifted 3 units in the positive
Functions p(x) and w(x) are graphed in the xy-plane. The x-direction and 3 units in the positive y-direction to obtain
graph of y = p(x) is equivalent to the graph of y = w(x) the graph of y = h(x). If f (x) = 4x + 10 and h(x) = ax + b,
translated 4 units upward and 3 units to the left, where the where a and b are real constants, what must be the value
positive x-direction is to the right and the positive y- of b?
direction is upward. Which of the following correctly relates
w(x) and p(x)?

a) w(x) = p(x – 3) + 4
b) w(x) = p(x + 3) + 4
c) w(x) = p(x – 3) – 4
d) w(x) = p(x + 3) – 4

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The Ivy 119 Dividing Polynomial, Function

5. April 2017 7.

In the xy-plane, the graph of y = (x – 6)2 + 3 is the image of y y = f (x)
the graph of y = (x + 5)2 + 3 after a translation of how many
units to the right? 6
5
4
3
2
1
– – –– – –– 123456 x

–
–
–
–
–

The accompanying figure shows the graph of y = f (x). If

function g is defined by g(x) = f (x + 4), then g (–1) could be

6. a) 3
y
b) 4

7236541 1 2 3 4 5 6 7 8 9 x c) 5

d) 6

–––– –––– –––– 8.

y

–––– 98
7654321
The graph of y = f (x + 5) is shown above. For which value f (x + 2)
of x, rounded to the nearest whole number, must f (x) = 0?
x
–––– –––– 123456789
––––

The graph of y = f (x + 2) is shown at left. For which value
of x must f (x) = 4?

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Dividing Polynomial, Function 120 The Ivy

9. y = f (x) x 11. December 2016
y
y 12345
5 y = f (x)
5 x
4 O5
3
2
1
–5 –4 –3 –2 –1–1

–2
–3
–4
–5

If n represents the number of different values of x for which The complete graph of the function f is shown in the xy-
f (x) = 2 and m represents the number of different values of x plane above. What is the y-intercept of the graph of
for which f (x) = 4, what is the value of mn? y = f (x – 2)?

a) –1
b) 0
c) 1
d) 2

10.

y
5
4
3 y = f (x)

2
1
–5 –4 –3 –2 –1 –1 1234 5x

–2
–3
–4
–5

Let m represent the number of points at which the graph of
y = f (x) and g(x) = 3 intersect. Let n represent the number of
points at which the graphs of y = f (x) – 1 and g(x) = 3
intersect. What is the value of m + n?

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The Ivy 121 Problem Solving and Data Analysis

Problem Solving and Data Analysis

Ratio, Proportion, Units

1. Ratios, rates, and proportions 3. June 2016
On Thursday, 240 adults and children attended a show. 1 palm = 4 fingers
The ratio of adults to children was 5 to 1.
How many children attended the show? 1 royal cubit = 7 palms
The equations above give the relationships between some
a) 40 ancient Egyptian units of length. A length of 28 fingers is
b) 48 equivalent to how many royal cubits?
c) 192
d) 200

2. Ratios, rates, and proportions 4. October 2016
On an architect’s drawing of the floor plan for a house, 1 On Tuesday, Dianna’s store sold 5 pairs of running shoes for
inch represents 3 feet. If a room is represented on the floor every 9 pairs of sandals sold. If Dianna’s store sold 27 pairs
plan by a rectangle that has sides of lengths 3.5 inches and 5 of sandals on Tuesday, how many pairs of running shoes did
inches, what is the actual floor area of the room in square Dianna’s store sell on that day?
feet?

a) 17.5
b) 51.0
c) 52.5
d) 157.5

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Problem Solving and Data Analysis 122 The Ivy

5. December 2016 8. Ratios, rates, and proportions
Isabel began charging her phone battery when it was 40% When David arrived in the country of Georgia, he
charged. After 48 minutes, her phone battery was 80% exchanged $165 of United States (US) currency for 385
charged. If the phone battery charges at a constant rate, how Georgian Laris. When he is ready to return to the US, he has
many more minutes will it take for her phone battery to be 147 Laris left. Assuming the exchange rate is the same, how
exactly 90% charged? many US dollars is that worth?

a) 4
b) 8
c) 10
d) 12

6. Ratios, rates, and proportions 9. October 2016
The rate of perceived exertion (RPE) scale runs from 0 to 20 A bucket contains 5 gallons of water. The water in the
and is approximately proportional to a person's heart rate. If bucket weighs 42 pounds. One gallon of diluted bleach is
an RPE rank of 20 corresponds to a person's maximum heart mixed with the water, and then the total weight of the liquid
rate of 205 beats per minute (BPM), what RPE rank in the bucket is 50.52 pounds. What is the weight per gallon
corresponds to the person's resting heart rate of 70 BPM? of the liquid in the bucket after the diluted bleach is added?
Round to the nearest integer. (Assume that the total volume of the liquid is the sum of
gallons of water and gallons of diluted bleach.)

a) 0.10 pounds/gallon
b) 0.12 pounds/gallon
c) 8.42 pounds/gallon
d) 8.50 pounds/gallon

7. Ratios, rates, and proportions

Miles traveled and gallons of fuel used

Mark Daniel 10. October 2016
The average net primary production in tropical rain forest
Gallons of fuel used 13 11 each year is 8,900 kilocalories per square meter. If the total
net primary production of a selected portion of a tropical
Miles traveled 286 M rain forest in a given year is 1.8108 kilocalories, what is
the approximate total area, in square meters, of the selected
The table above shows the number of gallons of fuel used by portion?

two drivers, Mark and Daniel, during two different car trips a) 4.9103
b) 1.6104
on the same highway. If the two drivers have the same c) 2.0104
d) 1.61013
model cars that get the same miles per gallon on the

highway then, to the nearest mile, m, how many miles was

Daniel's trip?

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The Ivy 123 Problem Solving and Data Analysis

11. Ratios, rates, and proportions 14. December 2016
Elena is conducting a study about the effects of toxins in the A cube made of osmium, the element with highest density,
water on the hormones of fish. Elena surveys 350 male fish measures 5 centimeters along each edge has a mass of 2825
in a river and finds that 150 of the male fish have egg cells grams. What is the density, in grams per cubic centimeter, of
growing inside them. According to Elena's survey, what is the osmium from which the cube is made? (Density is mass
the ratio of male fish with egg cells to male fish without egg per unit volume.)
cells in the river?

a) 3 : 4
b) 3 : 7
c) 4 : 5
d) 4 : 7

12. December 2016 15. April 2017
A recipe recommends cooking a roast between 35 and 45 A greenhouse owner purchases fertilizer in 60-gallon drums.
minutes per pound. Which of the following cooking times The fertilizer is mixed with water to make several batches of
for a 10-pound roast follows the recommendation? solution. Each batch of solution is made by mixing 3 quarts
of fertilizer with water. What is the maximum number of
a) 100 minutes batches of solution that can be made from one 60-gallon
b) 300 minutes drum of fertilizer? (1 gallon = 4 quarts)
c) 400 minutes
d) 500 minutes a) 45
b) 60
c) 80
d) 180

13. December 2016 16. May 2017
A dairy farm produces 5300 pounds of whole milk each day. At her summer job, Paula earns the same amount of money
The farm uses 21.2 pounds of whole milk to produce one for each hour she works. If she earns $240 for working 20
pound of butter. At this rate, how many pounds of butter can hours, how much does she earn for 5 hours?
the dairy farm produce each day?

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Problem Solving and Data Analysis 124 The Ivy

Questions 17 refer to the following information. 19. April 2017
During a storm, the atmospheric pressure in a certain
Most Frequently Reported Prior Career Fields location fell at a constant rate of 3.4 millibars (mb) per hour
over a 24-hour time period. Which of the following is
of the 535 Members of the 133th Congress closest to the total drop in atmospheric pressure, in
millimeters of mercury (mm Hg), over the course of 5 hours
Chamber during the 24-hour time period? (Note: 1,013 mb = 760 mm
Hg)
House of Senate
Representatives a) 2.6
b) 12.8
Business 187 27 c) 17.0
d) 22.7
Education 77 15

Law 154 55

Public service 180 42

The table above gives the distribution, by chamber, of fields

in which members of the 113th Congress of the United States

most frequently reported spending a portion of their careers

before being elected to Congress. There were 435 members

in the House of Representatives and 100 members in the

Senate at the time these data were collected. Some members Questions 20 refer to the following information.

reported more than one career field. Nutritional Information for 1-Ounce Serving of Seeds and

17. December 2016 Nuts

Of the 435 people who reported from the House of Seed or nut Calories Total fat Protein
(grams) (grams)
Representatives, 78 are female. Assuming that gender and

prior career field are independent, how many female Pecan 198 20.2 3.0

representatives would be expected to have reported a prior Pistachio 80 6.5 3.0

career field in public service? (Round your answer to the Pumpkin 159 13.9 8.5

nearest whole number.) Sunflower 166 14.6 5.9

Walnut 185 18.5 4.3

The table above shows the calories, grams of fat, and grams

of protein in 1-ounce servings of selected seeds and nuts.

20. April 2017

How many more grams of protein are in one pound of

18. November 2017 pumpkin seeds than are in one pound of pistachios?
A certain cake recipe calls for 3 cups of flour to make 12
servings. If 40 servings of the cake are being made, how (1 pound = 16 ounces)
many cups of flour will be needed for this recipe?
a) 48

b) 72

c) 88

d) 136

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The Ivy 125 Problem Solving and Data Analysis

21. November 2017 23. October 2017 US

The efficiency of an engine is the proportion of its fuel Washington High School randomly selected freshman,

energy that the engine can convert into motion energy. sophomore, junior, and senior students for a survey about
x
The equation E = 1 – p relates an engine’s efficiency, E, potential changes to next year's schedule. Of students
1 1
to its exhaust temperature x, in kelvins, and its operating selected for the survey, 4 were freshmen and 3 were

temperature p, in kelvins. A particular automobile engine sophomores. Half of the remaining selected students were

has an operating temperature of 473 kelvins and an exhaust juniors. If 336 students were selected for the survey, how

temperature of 293 kelvins. Based on the engine’s many were seniors?

efficiency, about how many joules of motion energy can the a) 240

engine obtain from 40,000,000 joules of fuel energy? b) 140

a) 15,200,000 c) 120

b) 24,800,000 d) 70

c) 64,600,000

d) 105,000,000

22. November 2017 24. December 2017
A doctor prescribes a medication for a 66-pound child. Water flows into a basin at a rate of 1 cubic foot per second.
In each dose, the child is to be given 8 milligrams of the At this rate, how many cubic feet of water will flow into the
medication for every kilogram of the child’s body weight. basin in one day?
The medication is to be given to the child 2 times per day for
5 days. How much medication, in milligrams, should be a) 1,440
provided for the entire 5-day treatment plan? b) 28,800
(1 kilogram=2.2 pounds) c) 86,400
d) 259,200

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Problem Solving and Data Analysis 126 The Ivy

25. December 2017 Questions 28 and 29 refer to the following information.
Each edge of a metal cube is 4 centimeters in length. The peregrine falcon can reach speeds of up to 200 miles per
If the metal cube has a mass of 172.8 grams, what is the hour while diving to catch prey, making it the fastest animal
density of the metal cube, in grams per cubic centimeter? on the planet when in a dive.

a) 2.7 28. October 2018 US
b) 10.8 What is the peregrine falcon’s maximum speed while diving
c) 14.4 to catch prey, in feet per second? (Round your answer to the
d) 43.2 nearest whole number. 1 mile = 5280 feet)

26. March 2018 29. October 2018 US
In pea plants, flowering time can be described as early, If a peregrine falcon dove at its maximum speed for half a
intermediate, or late. A cross of an early-flowering plant and mile to catch prey, how many seconds would the dive take?
a late-flowering plant results in a set of offspring having a (Round your answer to the nearest second.)
1:2:1 ration of early-, intermediate-, and late-flowering
plants. Plants, respectively. If 371 offspring are produced 30. May 2018
from a cross of an early-flowering plant and a late-flowering The circumference of Earth is estimated to be 40030
plant, the expected number of intermediate-flowering kilometers at the equator. Which of the following best
offspring would be closest to which of the following? approximates the diameter, in miles, of Earth’s equator?

a) 93 (1 kilometer  0.62137 miles)
b) 186
c) 247 a) 3205 miles
d) 278 b) 5541 miles
c) 7917 miles
27. October 2018 d) 13004 miles
The distance a certain object travels is proportional to the
amount of time the object is in motion. If the object travels
60 feet in 20 seconds, how many feet does the object travel
in 90 seconds?

a) 30
b) 50
c) 130
d) 270

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The Ivy 127 Problem Solving and Data Analysis

31. April 2019 US 33. March 2019
A larger version of the same map of the business district is
Selena created a scale model of an airplane where 1 printed with the side length 50% longer than the side length
of the previous map. On the larger map, what is the actual
centimeter on the model equals 6 meters on the airplane. distance, in feet, represented by 1 inch?

The wingspan of the model is 10.7 centimeters. Selena 34. Oct 2016 US
543210
wants to make a new model where a scale of 1 centimeter on out out out out out out Total
of 5 of 5 of 5 of 5 of 5 of 5
the model equals 3 meters on the airplane. Which of the
Day 1 2 3 4 6 2 3 20
following best describes how the wingspan of the new Day 2 2 3 5 5 4 1 20
Day 3 3 3 4 5 3 2 20
model will compare to the wingspan of the first model? ToTal 7 9 13 16 9 6 60
The same 20 contestants, on each of 3 days, answered 5
a) The wingspan of the new model will be 3 questions in order to win a prize. Each contestant received 1
point for each correct answer. The number of contestants
centimeters shorter than the first model. receiving a given score on each day is shown in the table
above.
b) The wingspan of the new model will be 3
No contestant received the same score on two different days.
centimeters longer than the first model. 1 If a contestant is selected at random, what is the probability
2 that the selected contestant received a score of 5 on Day 2 or
c) The wingspan of the new model will be as long Day 3, given that the contestant received a score of 5 on one
of the three days?
as the wingspan of the first model.

d) The wingspan of the new model will be 2 times as

long as the wingspan of the first model.

Questions 32 and 33 refer to the following information.
A town's chamber of commerce printed a map of the
business district. The map is 12 inches square, and 1 inch on
the map represents an actual distance of 300 feet.

32. March 2019
On the map, a parking lot for one business is represented by
a square with side length 0.25 inches. What is the actual
area, in square feet, of the parking lot?

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Problem Solving and Data Analysis 128 The Ivy

Questions 35 and 36 refer to the following information. 35. May 2019
According to Kosumi’s measurements, how many miles is it
Kosumi, located in the city of Redfield, South Dakota, is from Redfield to Sturgis?

trying to estimate the distances from Redfield to other cities 36. May 2019
Kosumi is planning a trip from Redfield to Pierre and
in the state. On a map, he measured the lengths along major expects to average 60 miles per hour while driving.
How long, in minutes, will it take for Kosumi to drive from
highways from Redfield to come other cities and listed these Redfield to Pierre?

lengths in the table below.

Distance from

City Redfield on map

(inches)

Chamberlain 2

Mitchell 1 3
Pierre 4
Sioux Falls 1
2 4

3

Sturgis 6

Watertown 1 1
2
3
The map that Kosumi used has a scale of 4 inches = 50

miles.

37. Table data

Enrollment trends in biologicallly focused programs

Bioinformatics Forensics Pre-Med Total

Male ____ ____ ____ 310

Female ____ ____ ____ 381

Total 97 214 380 691

A biology professor is investigating trends in current enrollment of male/female undergraduate students in certain biologically

focused cross-departmental programs. The above table represents the professor's findings. Note that students are not allowed to

enroll in more than one of these programs at this particular school. About how many female students should be enrolled in the

forensics program to provide evidence that enrolling in forensics is independent of student gender?

a) 66

b) 118

c) 210

d) 214

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The Ivy 129 Problem Solving and Data Analysis

Percent

Type 1: Finding a percent of a given number.

1. An estate agent charges 2% commission on every 5. A car valued at $6,500 last year is now worth 15%
house he sells. How much commission will he earn on less. What is its value now?
a house that he sells for $120,500?

2. At a Paris railway station, in one week 350 trains 6. A large factory employed 640 people. It had to
arrived. Of these trains, 5% arrived early and 13% streamline its workforce and lose 30% of the workers.
arrived late. How many arrived on time? How big is the workforce now?

3. Azwan, who was on a salary of $27,500, was given a 7. A computer system was priced at $1000 at the start of
pay rise of 7%. What is his new salary? 2008. At the start of 2009, it was 10% cheaper. At the
start of 2010, it was 15% cheaper than the price at the
4. In 2005 the population of a city was 1,565,000. By start of 2009. What is the price of the computer system
2010 it had increased by 8%. What was the population at the start of 2010?
of the city in 2010?

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Problem Solving and Data Analysis 130 The Ivy

8. In one year, the price of one share of ABC stock 11. What single discount is equivalent to two successive
increased by 20% in Quarter I, increased by 25% in discounts of 10% and 15%?
Quarter II, decreased by 20% in Quarter III, and a) 25%
increased by 10% in Quarter IV. By what percent did b) 24.5%
the price of the stock increase for the whole year? c) 24%
d) 23.5%

9. If the population of Town B is 50% greater than the 12. If the length of a rectangle is increased by 20% and the
population of Town A, and the population of Town C width is increased by 30%, then by what percent is the
is 20% greater than the population of Town A, then area of the rectangle increased?
what percent greater is the population of Town B than a) 50%
the population of Town C? b) 56%
a) 20% c) 65%
b) 25% d) It cannot be determined from the given
c) 30% information.
d) 35%

10. If a is 30% greater than A and b is 20% greater than B, 13. If increasing 20 by P percent gives the same result as
then ab is what percent greater than AB? decreasing 60 by P percent, what is P percent of 70?
a) 25%
b) 50%
c) 56%
d) 60%

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The Ivy 131 Problem Solving and Data Analysis

Type 2: Finding a number when a percent of it is given by using multiplier method.

1. On a grueling army training session, only 28 4. I received $385 back from the government, which
youngsters survived the whole day. This represented represented the 17.5% purchase tax on a piece of
35% of the original group. How large was the original equipment. How much did I pay for this equipment in
group? the first place?

2. Sales tax is added to goods and services. With sales tax 5. When a suit is sold in a shop the selling price is $171
at 17.5%, what is the pre-sales tax price of the and the profit is 80%. What was the cost price?
following priced goods?
T-shirt $9.87

Tights $1.41

Sweater $12.62 6. The cost of a pack of batteries, after a 5% tax, is $8.40.
What was the price before tax?
3. The number of workers in a factory fell by 5% to 228. a) $5.60
How many workers were there originally? b) $7.98
c) $8.00
d) $8.35

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Problem Solving and Data Analysis 132 The Ivy

7. What was the original price of a radio that sold for $70 10. The price of a stock falls 25%. By what percent of the
during a 20%-off sale?
a) $56 new price must the stock price rise in order to reach its
b) $84
c) $87.50 original value?
d) $90
a) 25%

b) 30%

c) 33 1 %
3
d) 40%

8. A stock decreases in value by 20 percent. By what 11. After a discount of 15%, the price of a shirt is $51.
percent must the stock price increase to reach its What was the original price of the shirt?
former value? a) $44.35
a) 15% b) $58.65
b) 20% c) $60.00
c) 25% d) $64.00
d) 30%

9. After getting a 20 percent discount, Jerry paid $100 for 12. After a 20% increase, the new price of a radio is
a bicycle. How much, in dollars, did the bicycle $78.00. What was the original price of the radio?
originally cost? a) $15.60
b) $60.00
c) $62.40
d) $65.00

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The Ivy 133 Problem Solving and Data Analysis

Type 3: Finding what percent one number is of another.

1. In 2009 a city raised $14,870,000 in local tax. In 2010 4. A certain brand of soup contains 10 percent tomato
it raised $15,597,000 in tax. What was the percentage
increase? paste by volume. Which of the following expressions

2. Before Anton started to diet, his mass was 95 kg. His represents the percent of tomato paste by volume in a
mass is now 78 kg. What percentage of his original
mass has he lost? mixture of s cups of soup and w cups of water?
10 s
a) w %

b) 10 %
s+ w
s+ w
c) 10 %

d) 10 s %
s+ w
s
e) 10( s+ w ) %

5. During course registration, 28 students enroll in a
certain college class. After three boys are dropped
from the class, 44% of the class consists of boys. What
percent of the original class did girls comprise?

3. What is the percent discount on a jacket marked down

from $120 to $100?
2 6. The price of a newspaper rises from 5 cents to 15
a) 16 3 % cents. What is the increase in price?
a) 50%
b) 20% b) 75%
c) 100%
c) 30% d) 150%
1 e) 200%
d) 33 3 %

e) 40%

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Problem Solving and Data Analysis 134 The Ivy

7. When Joe started his exercise program, he could do 12 10. On a test consisting of 80 questions, Marie answered
pushups in a minute. After he had been practicing for 75% of the first 60 questions correctly. What percent
two weeks, he could do 21 pushups in a minute. By of the other 20 questions did she need to answer
what percent did Joe’s rate of doing pushups increase? correctly for her grade on the entire exam to be 80%?
a) 9% a) 87.5%
b) 21% b) 90%
c) 42% c) 95%
d) 57% d) 100%
e) 75%

8. If exactly 50 out of the 400 seniors at a college are 11. Bob took 20 math tests last year. If he failed 6 of them,
majoring in economics, what percent of the seniors are
NOT majoring in economics? what percent of the math tests did he pass?
1
a) 37 2 %

b) 60%
1
c) 62 2 %

d) 66 2 %
3
e) 70%

9. A store sells a watch for a profit of 25 percent of the 12. After 2 months on a diet, John’s weight dropped from
wholesale cost. What percent of the selling price of the
watch is the store’s profit? 168 pounds to 147 pounds. By what percent did John’s
a) 12.5%
b) 20% weight drop?
c) 25% 1
d) 50% a) 12 2 %

b) 14 2 %
7
c) 21%

d) 25%
4
e) 28 7 %

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The Ivy 135 Problem Solving and Data Analysis

Interpreting Relationships Presented in Tables

1. October 2016 3. May 2018

Distribution of 250 History Books Roof type

Book type US British Asphalt Slate Cedar Total
shingle shake
Paperbacks 40 45

Hardcovers 90 75 Single story 9 4 2 15

A historian has a collection of 250 books about US and Two story 20 10 3 33

British history. The distribution of the books is shown in the Total 29 14 5 48

table above. If a hardcover book is to be selected at random, The table above shows the distribution of single-story and

what is the probability that the book will be a US history two-story houses in a neighborhood classified according to

book? 6 roof type. If one of the houses is selected at random, what is
11
a) the probability that it will be a single-story house with a slat

b) 13 roof? 4
25 48
4 a)
c) 9
b) 4
d) 9 15
25 4
c) 14

d) 14
48

2. April 2019 US 4. October 2017 US

Female Male Total Blood type

Blue eyes 2 4 6 Rhesus factor A B AB O

Brown eyes 8 6 14 + 33 9 3 37

Green eyes 1 5 6 – 721x

Total 11 15 26 Human blood can be classified into four common blood

Sierra recorded the gender and eye color of all the students types-A, B, AB, and O. It is also characterized by the

in her biology class. The results are shown in the table presence (+) or absence (–) of the rhesus factor. The table

above. If a male student is selected at random from Sierra’s above shows the distribution of blood type and rhesus factor

biology class, what is the probability that he will have brown for a group of people. If one of these people who is rhesus

eyes? 2 negative (–) is chosen at random, the probability that the
3
a) person has 1 blood type B is 1 . What is the value of x?
9
b) 2
5
3
c) 7

d) 3
13

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Problem Solving and Data Analysis 136 The Ivy

5. October 2018 7. April 2017

Grade Michael performed an experiment where he tossed two

9 10 11 12 Total coins, one dime and one nickel, at the same time and

Male 240 200 190 x recorded whether each one landed on heads or tails. His

Female 170 180 220 results are shown in the table below.

Total 5 x 350 410 Nickel
9
Heads Tails
The partially completed table above shows student
Heads 27 26
enrollment at Polk High School in January 2012. If a female Dime Tails 24 23

student is chosen at random, what is the probability that she For what percent of the tosses did the dime Michael tossed

is in grade 9? land on heads?

a) 47%

b) 49%

c) 51%

d) 53%

6. December 2016

Native continent 8. April 2017
Century and Region of United States
Class Australia Africa Presidents’ Birth as of 2014
Century
Mammals 8 6 18th 19th 20th
Northeast 5 6 3
Reptiles 9 17 South 9 4 3
Midwest 0 9 2
Birds 27 23 West 0 0 2

The table above shows the numbers of animals classified by The table above shows the distribution of United States
presidents according to the century and the region of the
class and native continent for all 90 animals at a local zoo. country in which they were born. Based on the information
in the table, what fraction of presidents who were not born in
What fraction of the reptiles are native to Australia? the nineteenth century were born in the South?
1
a) 10

b) 9
44
13
c) 45

d) 9
26

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The Ivy 137 Problem Solving and Data Analysis

9. May 2017 US 11. May 2019

Treatment Number of plants Car component Not Defective Total
Regressed Thrived Total defective

A 80 120 200 Component A 225 25 250

B 140 60 200 Component B 440 10 450

The table above shows the result of an experiment involving Component C 285 15 300

the effect of two treatments, A and B, on plants. Based on Total 950 50 1000

the results, what fraction of the plants that thrived received The table above summarizes the results of testing 1000 car

treatment A? components of three different types to determine whether
2
a) 5 they were defective. Of the defective components, what

b) 1 fraction were component B?
2
3
c) 5

d) 2
3

10. March 2019 12. March 2018 US

Purchase Rent Number of Flight Arrivals at
equipment equipment
Total Centerville Airport in a Month

Floor sander 347 211 558 On time Delayed Total

Power washer 286 134 420 Airline A 2,029 861 2,890

Total 633 345 978 Airline B 1,150 700 1,850

The table above shows the numbers of customers who have Total 3,179 1,561 4,470

purchased or rented either a floor sander or a power washer Based on the table above, what fraction of the flights for

from a home-improvement store. Based on the table, what Airline A were delayed?
700
proportion of the floor-sander customers purchased their a) 1,850

equipment? b) 861
1,561
347 861
a) 633 c) 2 ,890

b) 558 d) 2 ,029
978 2 ,890
347
c) 558

d) 633
978

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Problem Solving and Data Analysis 138 The Ivy

Questions 13 and 14 refer to the following information. 15. May 2018 US

Number of pets Homeroom teacher The table below shows the number of lakes in the United

Ms. Wilcox Mr. Xiong Kingdom classified by alkalinity and depth.

None 2 3 Depth class Alkalinity class

One 10 8 Low Medium High Total

Two 8 9 Shallow 87 61 209 357

Three 7 4 Moderate 227 86 110 423

Four or more 3 6 Deep 130 35 21 186

Teachers in two homerooms of 30 students each surveyed Total 444 182 340 966

their students to find out the number of pets the students If a lake has high alkalinity, which of the following is closest

have. The results of the survey are shown in the table above. to the probability that the lake also has a shallow depth?

13. March 2018 a) 0.22

What fraction of the 60 students indicated having four or b) 0.37

more pets? c) 0.59

d) 0.61

16. October 2017 US

Customer Purchases at a Gas Station

Beverage Beverage not Total
purchased purchased

Gasoline 60 25 85
purchased
14. March 2018
If a student in Ms. Wilcox’s homeroom who indicated Gasoline not 35 15 50
having one or more pets is selected at random, what is the purchased
probability that the selected student indicated having exactly
three pets? Total 95 40 135

On Tuesday, a local gas station had 135 customers. The

table above summarizes whether or not the customers on

Tuesday purchased gasoline, a beverage, both, or neither.

Based on the data in the table, what is the probability that a

gas station customer selected at random on that day did not

purchase gasoline?
15
a) 50

b) 15
40
35
c) 50

d) 50
135

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The Ivy 139 Problem Solving and Data Analysis

17. October 2016 Based on the table, what proportion of students who majored
in political science did not have a career in law, medicine, or
Majors Career teaching directly after college?
Law Medicine Teaching Other Total
a) 0.04
English 23 5 32 30 90 b) 0.12
c) 0.16
Biology 2 78 20 10 110 d) 0.20

Political 54 1 9 16 80
Science

Economics 13 10 17 80 120

Total 92 94 78 136 400

The table above shows the number of students who were

enrolled in one of four majors when they were in college and

the career that each student chose directly after college.

18. March 2019 US

Observed Matings among Fruit Flies

Female fruit fly group

Female raised on Female raised on Total
starch maltose

Male fruit fly group Male raised on starch 22 9 31
Male raised on maltose 8 20 28

Total 30 29 59

The table above shows the observed mating frequencies among a group of fruit flies raised on either a starch medium or a maltose

medium. What fraction of the observed matings were between fruits flies that were raised on the same medium?
9
a) 31

b) 17
59
31
c) 59

d) 42
59

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Problem Solving and Data Analysis 140 The Ivy

19. October 2018 US

Survey Results

Answer Percent

Never 31.3%

Rarely 24.3%

Often 13.5%

Always 30.9%

The table above shows the results of a survey in which tablet users were asked how often they would watch video advertisements in

order to access streaming content for free. Based on the table, which of the following is closest to the probability that a tablet user

answered “Always,” given that the tablet user did not answer “Never”?

a) 0.31

b) 0.38

c) 0.45

d) 0.69

20. Table data

Route A On time Late Total
Route B 5 6 11
Route C 2 10 12
Total 6 11 17
13 27 40

Victor decides to try three different routes to work for a period of 40 days. In the table above, he tracked whether he arrived to
work late or on time each time that he used a particular route. According to the table, what is the probability that Victor was late
given that he used Route A?

a) 3 b) 2
20 9

c) 3 d) 6
5 11

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The Ivy 141 Problem Solving and Data Analysis

Interpreting Relationships Presented in Scatterplots

Number of pints sold 1. Scatterplots y = 233 – 32x A grocery store sells pints of raspberries and sets the price per
pint ach week. The scatterplot above shows the price and the
111111111197832864095170000000000000 number of pints of raspberries sold for 19 weeks, along with
0 the line of best fit and the equation for the line of best fit.

Price per pint (dollars)

a. According to the line of best fit, how many pints of b. For how many of the 19 weeks shown was the number
raspberries would the grocery store expect to sell of pints of raspberries sold greater than the amount
in a week when the price of raspberries is $4.50 predicted by the line of best fit?
per pint?

Num rbeetruorfnesdurveys2. May 2017 US The scatterplot above shows the number of surveys sent to and
returned from people in 17 different neighborhoods. A line of best
Return eSdurfvroemys 1S7enNt etiogahnbdor hoods fit for the data is also shown. For the neighborhood that had surveys
876500000000 sent to 800 people, which of the following is closest to the positive
43210000000000 20N0um4b0e0r o6f0s0urv8ey00s s1e,n0t00 difference between the actual number of surveys returned and the
number predicted by the line of best fit shown?

a) 150
b) 170
c) 200
d) 250

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Problem Solving and Data Analysis 142 The Ivy

Questions 3 and 4 refer to the following information.

31
3290
Average fuel economy 28
(miles per gallon) 27
26
25
24
23
22
21 0 10 20 30 40 50 60 70 80

Speed (miles per hour)

The scatterplot above shows the average fuel economy for a certain class of car driven at 12 different speeds. The graph of a

quadratic model for the data is also shown.

3. May 2017 US 4. May 2017 US

For what fraction of the 12 speeds does the model The quadratic model predicts the average fuel economy to

overestimate the average fuel economy? be 26 miles per gallon for how many different speeds?

Number of tulip bulbs5. March 2019 6. May 2019
Annual visitors40 Railroad Museum Visitors

30 80,000

20 60,000

10 40,000

0 10 20 30 40 50 60 20,000
Number of daffodil bulbs
0 5 10 15
Alice is shopping at a garden center where tulip bulbs cost Years since 1968
S1. 50 each and daffodil bulbs cost $0.50 each. Each point
on the graph above represents one combination of tulip and The scatterplot above shows the number of visitors to a
daffodil bulbs that Alice can buy using her entire gardening railroad museum in Pennsylvania each year from 1968 to
budget. Based on the graph, if Alice spends her entire budget 1980, where t is the number of years since 1968 and n is the
on daffodil bulbs, how many daffodil bulbs does she buy? number of visitors. A line of best fit is also shown. Which of
the following could be an equation of the line of best fit
a) 20 shown?
b) 30
c) 60 a) n = 16,090 + 4,680t
d) 80 b) n = 4,690 + 16,090t
c) n = 16,090 + 9,060t
d) n = 9,060 + 16,090t

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The Ivy 143 Problem Solving and Data Analysis

Questions 7 and 8 refer to the following information.

Number of tickets sold 25 Demand for Concert Tickets
(thousands)
20

15

10

540 50 60 70 80 90
Price per ticket (dollars)

The scatterplot above shows the relationship between the number of tickets sold and the price per ticket for 9 different concerts.

The line of best fit is also shown.

7. December 2016 8. December 2016

Which of the following is closest to the mean number of For a concert where the ticket price was $100, the line of

tickets sold for the 9 concerts shown in the scatterplot? best fit correctly predicted the number of tickets sold. How

a) 20,000 many tickets were sold for that concert?

b) 17,000 a) 6,000

c) 15,000 b) 8,000

d) 13,000 c) 10,000

d) 12,000

Longevity (years)9. March 2019 The scatterplot above shows the average heart rate and
Average Heart Rate versus Longevity longevity for 7 different types of animals. A line of best fit for
the data is also shown. According to the line of best fit, which
90 of the following best estimates the decrease in longevity, in
80 years, for each increase of 10 beats per minute of heart rate?
70
60 a) 1
50 b) 12
40 c) 24
30 d) 32
20
10
0 0 10 20 30 40 50 60 70 80

Heart rate (beats per minute)

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Problem Solving and Data Analysis 144 The Ivy

10. March 2019 Actual Voters versus The scatterplot above shows the percent of residents who were
70% Registered Voters registered to vote, x, and the percent of residents who actually
voted, y, on election day in 2010, for each of the 50 states in
Percent of actual voters 60% the United States. Which of the following could be the slope
of a line of best fit for the data?
50%
a) –6.4
40% b) 0.8
c) 6.4
d) 8.8

30%
30% 40% 50% 60% 70%
Percent of registered voters

Speed (mph)11. April 2019 US The scatterplot above shows the maximum height h, in feet
130 MaximumoSfp1e2edRvoelrlesrusCMoaasxteimrsum Height (ft), and maximum speed s, in miles per hour (mph), of 12
120 roller coasters as well as the line of best fit for the data. Of the
110 following, which best represents an equation for the line of
100 best fit?
90
80 a) s = 0.21h+ 32
70 b) s = 0.43h+ 32
60140 180 220 260 300 340 380 420 c) s = 0.21h+ 62
Height (ft) d) s = 0.43h+ 62

(milGeass pmeirlegaalgleon)12. June 2016 The scatterplot above shows the weight and gas mileage of fourteen
Gas Mileage versus cars. Which of the following can be concluded from the data in the
scatterplot?
50 Weight of Fourteen Cars
40 a) If the weight of car A is greater than the weight of car B, then
30 the gas mileage of car A is greater than the gas mileage of car
B.
20
10 b) If the weight of car A is greater than the gas mileage of car B,
20,000 2,500 3,000 3,500 4,000 then the gas mileage of car A is less than the gas mileage of
car B.
Weight of car (pounds)
c) A line of best fit for the data has a positive slope.
d) A line of best fit for the data has a negative slope.

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The Ivy Maximum temperature (°F) 145 Problem Solving and Data Analysis

13. March 2018 The scatterplot above shows the minimum and maximum temperatures
Minimum and Maximum in July for 19 selected cities in the United States. For the cities shown,
which of the following is closest to the maximum temperature for the
Temperatures in July for Selected Cities city with the median minimum temperature?

111710099887665505005005045 50Mini5m5um6t0empe6r5atur7e0(°F)75 80 a) 64 °F
b) 66 °F
c) 84 °F
d) 86 °F

Total fat (g) 14. March 2018 US The scatterplot above shows the relationship between the amount
ChTooletaslteFraotl aonfdSDaniedtwaricyhes of dietary cholesterol, in milligrams (mg), and the amount of
Number of home runs total fat, in grams (g), in the 12 sandwiches offered by a certain
85 restaurant. The line of best fit predicts the amount of total fat a
80 sandwich has based on the amount of dietary cholesterol in the
75 sandwich. How many grams of total fat are in the sandwich for
70 which this prediction is the most accurate?
60
60 a) 140
55 b) 115
50 c) 85
45 80 100Die1t2a0ry 1ch40ole1st6e0rol1(t8m0g)200 220 d) 60

15. May 2018 The scatterplot above shows the total number of home runs hit in
major league baseball, in ten-year intervals, for selected years.
Total Home Runs for The line of best fit for the data is also shown. Which of the
15443612253,,,,,,,,,,,5005550500050000000000000000000000000 Selected Years 1903-2013 following is closest to the difference between the actual number
of home runs and the number predicted by the line of best fit in
Year 2003?

a) 250
b) 500
c) 750
d) 850

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Problem Solving and Data Analysis Revenue (millions of dollars) 146 The Ivy

16. May 2018 The scatterplot above shows the revenue, in millions of dollars,
that a company earned over several years and a line of best fit for
70 Revenue the data. In Year 4, the difference between the actual revenue and
60 the predicted revenue is n million dollars, where n is a positive
50 integer. What is the value of n? Round your answer to the nearest
40 whole number. (Disregard the $ sign when gridding your
30 answer.)
20
10
00 1 2 3 4 5 6 7

Year

Average production cost17. May 2018 US The scatterplot above shows the average production cost, in cents
(cents per pound) 100 per pound, of coffee in Ecuador for the years from 2002 to 2012.
90 A line of best fit is also drawn. Which of the following is closest
80 2005 2010 to the difference, in cents per pound, between the actual average
70 Year production cost in 2012 and the average production cost in 2012
60 predicted by the given line of best fit?
50
40 a) 4
30 b) 8
20 2015 c) 16
10 d) 50
0
2000

18. October 2018

Annual sales (millions of dollars) 14 The scatterplot above shows the relationship between the
square footage of 14 clothing stores and their annual sales, in
12 millions of dollars. Based on the line of best fit (not shown),
10
8 which of the following is the best prediction for amount of
6 annual sales, in millions of dollars, a clothing store with 4
4 thousand square feet would earn?

2 a) 4.1

00 1 2 3 4 5 6 7 b) 6.2
Square footage of store c) 7.8
(thousands of square feet) d) 10.1

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