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The Ivy 47 Heart of Algebra

29. 31. October 2019
3(x + 4) = 2(x + c) + x
3y = 6x – 6
2x = 6y – N In the equation shown, c is a constant. For what value of c
In the system of equations above, N is a constant. For which does the equation have infinitely many solutions?
value of N is there exactly one solution (x, y) with y = 2?
a) 2 a) 4
b) 8 b) 6
c) Any real number c) 8
d) No real number d) 12

30.

2x + by = 10
y = 2x – 2

In the system of equations above, b is a constant. For which
value of b is there exactly one solution (x, y) with y = 2?

a) 3
b) 6
c) 7
d) None of the above

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Heart of Algebra 48 The Ivy

Linear Inequality

1. 3.
4 – 2x  1 – x
Which of the following best describes the solutions to the 7x + 1 < x + 9
inequality shown above?
a) x  1 Which of the following best describes the solutions to the
b) x  3
c) x  3 inequality shown above?
d) x  –5 4
a) x > 3

b) x < 4
3
c) x < 1
5
d) x < 3

2. 4.

6x – 2 < 2x + 3 52 – 3x < –14

Which of the following best describes the solutions to the Which of the following best describes the solutions to the

inequality shown above? inequality shown above?
38
a) x < 5 a) x < − 3
4
5 38
b) x > 4 b) x > 3

c) x < 1 c) x < 22
4
5 d) x > 22
8
d) x <

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The Ivy 49 Heart of Algebra

5. 8.
If 9 ≥ 4x + 1, which inequality represents the possible range
8x + 6 < 4x + 10 of values of –12x – 3?

Which of the following best describes the solutions to the a) –12x – 3 ≥ –17
b) –12x – 3 ≤ –17
inequality shown above? c) –12x – 3 ≥ –27
1 d) –12x – 3 ≤ –27
a) x < 3

b) x < 4

c) x < 1

d) x > 1

6. 9.
If –18  6 – 12ax, which inequality represents the possible
range of values of 1 – 2ax? If 15bx – 20 > 35, where b is a positive constant, which of

a) 1 – 2ax  –23 the following best describes all possible values of 4 – 3bx?
b) 1 – 2ax  –23
c) 1 – 2ax  –3 a) Any value less than –7
d) 1 – 2ax  –3
b) Any value greater than –7
11
c) Any value less than – 3b

d) Any value greater than 11
3b

7. 10. 3 16
If 9 ≥ 4x + 1, which inequality represents the possible range 5 5
of values of 12x + 3? If –8 < − m + 1 ≤ − , what is one possible value of m?

a) 12x + 3 ≥ 17
b) 12x + 3 ≤ 17
c) 12x + 3 ≥ 27
d) 12x + 3 ≤ 27

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Heart of Algebra 50 The Ivy

11. 12.
If –3x + 6 ≥ 18, which of the following must be true?
4c + 5 < 4c + 3
a) x ≤ –4
b) x ≤ 8 Which of the following best describes the solutions to the
c) x ≥ –4
d) x ≥ –8 inequality shown above?

a) All real numbers
1
b) c < 2

c) c > 1
4
d) No solution

Absolute Value

1. 2.
|x + 3| = 6 If 5 < |k + 1| < 6 and k < 0, then what is one possible value of
|y – 2| = 7 |k|?
For the equation shown above, which of the following is a
possible value of x – y?
a) –14
b) –4
c) –2
d) 14

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The Ivy 51 Heart of Algebra

3. May 2018 5.
If |2x + 3| = 5 and |3y – 3| = 6, what is one possible value of
|xy|? A company sells precision glass rods that are 100 mm in
length for $15 each. If a glass rod is not 100 mm in length,
the company offers a $0.75 discount for each 1 mm that the
rod is either too long or too short. Which of the following
expressions can be used to compute the price of a rod that is
x mm in length?

a) 15 + 0.75|x – 100|
b) 15 – 0.75|x – 100|
c) 15 – 0.75|x + 100|
d) 15 + 0.75|x + 100|

4. 6.

One of the rules in a public speaking contest requires The stratosphere is the layer of the Earth’s atmosphere that
is more than 10 kilometers (km) and less than 50 km above
contestants to speak for as close to 5 minutes (300 seconds) the Earth’s surface. Which of the following inequalities
describes all possible heights x, in km, above the Earth’s
as possible. Contestants lose 3 points for each second they surface that are in the stratosphere?

speak either over or under 5 minutes. Which expression below a) |x + 10| < 50
b) |x – 10| < 50
can be used to determine the number of points a contestant c) |x + 30| < 20
d) |x – 30| < 20
loses if she speaks for x seconds?

a) 3|x – 300|

b) 5|x – 300|

c) 3|x + 300|
3
d) 5 |x + 300|

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Heart of Algebra 52 The Ivy

7. 9.

The minimum speed on a given stretch of freeway is 45 mph, A lumber company producing 13-inch boards can only sell
and the maximum speed is 65 mph. Which of the following 13 116
inequalities could be used to decided if a motorist driving at a ones cut between 12 15 and . If it sells a board that is
rate of r mph is within legal limits? 16
x inches long, which of the following describes all possible
a) |r – 45| ≤ 10
b) |r – 55| ≤ 10 values of x? 1
c) |r – 65| ≤ 10 16
d) |r – 55| < 10 a) x −13 = 1
b) x −13  16

c) x +13  1
16
1
d) x −13  16

10.
If |x – 16|  4 and |y + 6|  2, what is the greatest possible
value of x – y?

8. 11.
1 1
In a certain greenhouse for plants, the Fahrenheit temperature, If 2 x  2 , then which statement must be true?
F, is controlled so that it does not vary from 79o by more than
7o. Which of the following best expresses the possible range
in Fahrenheit temperatures of the greenhouse?

a) |F – 79|  7
b) |F – 79| > 7
c) |F – 7|  79
d) |F – 7 | > 79

a) x ≤ –2 or x ≥ 2

b) x ≤ –1 or x ≥ 1
1 1
c) x  − 2 or x 2

d) –1 ≤ x ≤ 1

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The Ivy 53 Heart of Algebra

12. March 2018 1 Question 15 refer to the following information
2
x+4 = 2 At regular checkups, a pediatrician measures Anne’s height

Which of the following values of x satisfies the equation and weight and reports each as a percentile for her age. For

above? example, when Anne was 24 months old, her weight

I. –12 percentile of 65 indicated that 65% of children that age were

at or below Anne’s weight. Anne’s results for 6 checkups

II. –4 are shown below.

III. 4 Checkup age Height percentile Weight percentile

a) II only 1 month 40 35

b) III only 3 months 45 45

c) I and II only 6 months 55 50

d) II and III only 12 months 60 55

18 months 55 55

24 months 50 65

13. 15. June 2016
If |x|  2 and |y|  1, then what is the least possible value of If p represents the height percentile value reported for Anne
x – y? during the 6 checkups, which of the following is true?

a) –3 a) |p| ≤ 10
b) –2 b) |p – 10| ≤ 5
c) –1 c) |p – 40| ≤ 10
d) 0 d) |p – 50| ≤ 10

14. 16. October 2019
Two different points on a number line are both 3 units from
the point with coordinate –4. The solution to which of the (t− 2)2 = t – 2
following equations gives the coordinates of both points? If a is a solution to the equation above, which of the
following must be true? (Oct 2019)
a) |x + 4| = 3
b) |x – 4| = 3 a) a ≥ 2
c) |x + 3| = 4 b) a = 2
d) |x – 3| = 4 c) –2 ≤ a ≤ 2
d) a ≤ –2

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The Ivy 54 Polynomial and Quadratic

Passport to Advanced Math

Operations with Polynomials and Rewriting Expressions

1. 4. March 2019
(x2 + bx – 2)(x + 3) = x3 + 6x2 + 7x – 6 Which of the following expressions is equivalent to
In the equation above, b is a constant. If the equation is true 2(x2 – 1) + 3(x2 + 2)?
for all values of x, what is the value of b?
a) 2 a) 2x2 – 6
b) 3 b) 5x2 + 3
c) 7 c) 5x2 + 4
d) 9 d) 6x2 – 2

2. April 2019 US 5. March 2019
Which of the following expressions is equivalent to: If (2x + 3)(x + 2) = ax2 + bx + c for all x, what is the value of
a + b + c?
(2x + 6) + (x2 + 2x + 1)
a) x2 + 5 6. March 2019
b) x2 + 7 x4 −16
c) 4x2 + 7 x−2
d) x2 + 4x + 7 Which of the following is equivalent to the expression
above, where x > 2?
3. April 2019 US a) x3 – 8
x4 – 8x2 + 16 b) x3 + 8
Which of the following is equivalent to the expression c) (x – 2)(x2 – 4)
above? d) (x + 2)(x2 + 4)
a) (x – 2)2(x + 2)2
b) (x2 + 4) (x + 2)(x – 2)
c) (x – 2)4
d) (x – 4)4

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Polynomial and Quadratic 55 The Ivy

7. 10.
Which of the following is equivalent to 16s4 – 4t2?
5 x  3
6 y 10 y a) 4(s2 – t)(s2 + t)
b) 4(4s2 – t)(s2 + t)
Which expression is equivalent to the above product for all c) 4(2s2 – t)(2s2 + t)
d) (8s2 – 2t)(8s2 + 2t)
y > 0? x
2
a)

b) 25 x
9
x
c) 2 y2

d) x
4 y2

8. March 2019 US 11.
Which of the following is equivalent to the sum of If 2x – 3y = 5, what is the value of 4x2 – 12xy + 9y2?
3x4 + 2x3 and 4x4 + 7x3?
a) 5
a) 16x14 b) 12
b) 7x8 + 9x6 c) 25
c) 12x4 + 14x3 d) 100
d) 7x4 + 9x3

9. 12. April 2019 US
5(x + a) + 3(x2 – a) = 3x2 + 5x + 4
l5 m4 n6
pq In the equation above, a is a constant. If the equation is true
for all values of x, what is the value of a?
l5 m3 n6
p2q2

Which expression is equivalent to the above quotient for all

m, n, p, q > 0?
1
a) mpq

b) mpq

c) l10 m7 n12
p3q3

d) 1

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The Ivy 56 Polynomial and Quadratic

13. October 2018 16.
Which of the following is equivalent to the expression –(x – 2)(x – 4)
(x + 2)3 – (x + 2)2? Which of the following expressions is equivalent to the
above expression?
a) x + 2 a) –x2 + 6x – 8
b) x3 – x2 + 4 b) –x2 – 2x – 8
c) x3 + 5x2 + 8x + 4 c) x2 – 6x + 8
d) x3 + 5x2 + 16x + 12 d) x2 + 2x – 8

14. 17.
Which of the following expressions is equivalent to (3y + 8y2 + 2) – 3(y2 – 2y + 5)
(3c – 4c2 + c3) – (5c2 + 8c3 – 6c)? Which of the following is equivalent to the above
expression?
a) 7c3 – 4c2 – 3c a) 5y2 + y + 7
b) 7c3 + c2 – 3c b) 5y2 + 5y – 17
c) –7c3 + c2 + 9c c) 5y2 – 3y + 17
d) –7c3 – 9c2 + 9c d) 5y2 + 9y – 13

18. October 2018

15. October 2018 US 2(x − 3 )(x + 4)
If u + t = 5 and u – t = 2, what is the value of 2
(u – t)(u2 – t2)?
Which of the following is equivalent to the expression

above?

a) x2 + 5x – 6

b) 2x2 + x – 6

c) 2x2 + 5x – 12

d) 2x2 – 12

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Polynomial and Quadratic 57 The Ivy

19. October 2018 US 22. March 2018
Which of the following is equivalent to 2x(x2 – 3x)? When 2(9x + 5y)(–11x + 3y) is rewritten in the form
ax2 – bxy + cy2, what is the value of b?
a) –4x2
b) 3x3 – x2
c) 2x3 – 3x
d) 2x3 – 6x2

20. 23. May 2018
(a2 + b)2 – (a2 – b)2 Which of the following equations is equivalent to
Which of the following is equivalent to the above x2 + 10x + 21?
expression?
a) –4a2b a) (x + 1)(x + 9) + 12
b) 2(2ab) b) (x + 1)(x + 9) + 12x
c) 4a2b c) (x + 3)(x + 7) + 5
d) b2 d) (x + 3)(x + 7) + 5x

21. 24. March 2018 US
2 1
If x2 = r2 + 2rs + s2, y2 = r2 – s2, x > 0, and y > 0, then y = 3x2 − 6x2
x
r+ s
a) r−s Which of the following expressions is equivalent to the

b) r−s expression above for x > 0?
r+ s 1
r−s a) − 2 x 2
rs
c) b) − 1
x
d) r2 + s2 3 2
r2 − s2
c) 1
3x2
1
d) 2x2

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The Ivy 58 Polynomial and Quadratic

25. 28. May 2018 US
(w – 3)5 – (w – 3)2 Which of the following is equivalent to (2x + 4)2 – 4x2?
Which of the following is equivalent to the above
expression? a) 16(x+ 1)
a) (w – 3)3 b) 8(x+ 2)
b) w5 – w2 – 252 c) 4(4x+ 1)
c) (w – 3)2(w – 3)3 d) 2(8x+ 1)
d) (w – 3)2((w – 3)3 – 1)

26.

8 c2 + 2 c3 29. May 2018 US
2 c2 −32 If 3x – 6y = 9z, which of the following expressions is
equivalent to x2 – 4xy + 4y2?
Which expression is equivalent to the above for all c > 1?
c a) 9z
a) − 4 b) 3z2
c) 9z2
b) c2 d) 81z2
c+ 4
c2
c) c−4

d) − c2
c−4

27. 30.

x3+9x2 49m4 n− 21m6 n2
x3 7m2 n4

Which expression is equivalent to the above for all x > 1? Which expression is equivalent to the above for all m > 1

a) 9x2 and n > 1?

b) 1 + 9x2 a) 7m2n4 – 3m4n2
x+9
c) x b) 7m2n3 – 3m4n2

d) 1+9x c) 7m2 −3m3
x n2
7m2−3m4 n
d) n3

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Polynomial and Quadratic 59 The Ivy

31. 34.
If (x + 5)(x + p) = x2 + 2x + k, then
5 m2 + 7m − 2 2m 9
2m−9 m− a) p = 3 and k = 5
9 b) p = – 3 and k = 15
Which expression is equivalent to the above for m  2 ? c) p = 3 and k = – 15
d) p = – 3 and k = – 15
a) 3m2 + 7m
2m−9
5m2 + 5m
b) 2m−9

c) 5m+ 9 m
2m−9
5m2 + 7 m−1
d) 2m−9

32. 35.
a2bc+ ab2c+ abc2
For all x   1 , 2x2+5x−3 =
abc 2 4 x2−1
If abc  0, Which expression is equivalent to the expression 2x−3
above? a) 2x+1

a) a + b + c b) x−3
b) a + b + cabc 2x+1
c) a3b3c3 2x+3
d) 3abc c) 2 x−1

d) x+3
2x+1

33. 36. a b
If (2x2 + 5x – 3)(4x – 1) = ax3 + bx2 + cx + d for all values of a2 − b2 a2 − b2
x, what is the value of b? The sum of and is

a) –22 a) 1
b) –18 a−b
c) 18 a
d) 22 b) a−b

c) b
a−b
a+ b
d) a−b

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The Ivy 60 Polynomial and Quadratic

37. 39.

If ax + x2 = y2 – ay, what is a in terms of x and y? If h = x2 −1 + x2 −1 , what is x in terms of h?
x+1 x−1
a) y – x h
a) 2
b) x – y

c) x + y b) 2h + 1

d) x2+ y2 c) 2h – 1
x+ y
d) h
2

38. a
b
If ax2 – bx = ay2 – by, then = 40. May 2018
Which of the following is an equivalent form of the
a) 1 expression (2x – 2)2 – (2x – 2)?
x− y
1 a) 2x2 – 6x + 6
b) x+ y b) 4x2 – 10x + 2
c) (2x – 2)(2x – 2
c) x− y d) (2x – 3)(2x – 2)
x+ y
x+ y
d) x− y

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Polynomial and Quadratic 61 The Ivy

Quadratic Equations

1. 3.
Which of the following could be the factored form of the Determine where f (x) intersects the x-axis.
polynomial shown below?
f (x) = (x + 6)2 – 16
(x + 8)2 – 1 = 0 a) x = –2, x = 10
a) x = –7, x = –9 b) x = 2, x = –10
b) x = –7, x = 9 c) x = –2, x = –10
c) x = 7, x = –9 d) f (x) does not intersect the x-axis.
d) There are no solutions.

2. 4.
Determine where f (x) intersects the x-axis. Solve for x:

f (x) = (x – 10)2 – 49 (x – 1)2 – 9 = 0
a) x = 17, x = 3 a) x = –4, x = –2
b) x = 17, x = –3 b) x = 4, x = 2
c) x = –17, x = 3 c) x = 4, x = –2
d) f (x) does not intersect the x-axis. d) There are no solutions.

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The Ivy 62 Polynomial and Quadratic

5. June 2016 8. March 2018
Which of the following equations has exactly one real
solution? 4x2 + 3x – 1 = 0

a) x2 – 6x + 9 = 0 What are the solutions to the equation above?
b) x2 – 5x + 6 = 0
c) x2 + 2x – 1 = 0 a) x = −3  5
d) x2 + 3x – 9 = 0 8
35
b) x = 8

c) x= −3 5
4
−3  13
d) x = 8

6. June 2016 9.
The square of n is equal to the sum of twice n and 24.
If n is positive, what is the value of n? 1,000 = 20z2
How many distinct real solutions does the equation above
have?

a) 0
b) 1
c) 2
d) 4

10. 1
3
(y – k)y =

In the equation above, k is a constant. One of the solutions to

7. March 2019 US the equation is:
If t > 0 and (3t)2 – 5(3t) – 14 = 0, what is the value of t?
3+ 9+ 4( 1 )
2 3

What is the value of k?

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Polynomial and Quadratic 63 The Ivy

11. 14.
–2x2 + 5x = 17 What is the product of all the solutions to the equation
Which of the following describes the solutions to the 3z2 – 12z + 6 = 0?
equation above?
a) The equation has two distinct real, rational a) 2
b) 2
solutions. c) 4
b) The equation has two distinct real, irrational d) 4 2

solutions.
c) The equation has one distinct real solution.
d) The equation has no real solutions.

12.

100 – 121k2 = 0 15. 13 15
2 2
What are the solutions to the equation above? x 2 + x + = 0

a) k = 100 Let x = a and x = b are the solutions to the equation above,
121
100 100 what is the value of ab?
b) k = − 121 and k = 121
a) –15
c) k = 10 15
11 b) − 2
10 10
d) k = − 11 and k = 11 c) 15
2
d) 15

13. 16. May 2019
x2 – ax + 12 = 0
What are the solutions x of x2 – 3 = x? In the equation above, a is a constant and a > 0. If the
equation has two integer solutions, what is a possible value
a) −1 11 of a?
2
−1 13
b) 2

c) 1 11
2
1 13
d) 2

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The Ivy 64 Polynomial and Quadratic

17. 20. April 2019 US
In the expression x2 + kx + 12, k is a negative integer. If 4x2 + bx + 9 = 0, where b is a constant, has exactly one
Which of the following is a possible value of k? solution, what is a possible value of b?

a) –13 a) 72
b) –12 b) 36
c) –6 c) 12
d) 7 d) 6

18. May 2019 US 21. Level 2 (Solving Quadratic Equations)

2x2 – 2 = 2x + 3 kx2 + 5x = –7

Which of the following is a solution to the equation above? In the equation above, k is a constant. Which of the

a) 2 following inequalities represents all values of k for which

b) 1 – 11 the equation has two distinct real solutions?
1 25
c) 2 + 11 a) k < − 28

d) 1+ 11 b) k > − 25
2 28
25
c) k < 28

d) k > 25
28

19. November 2017

x2 – 6x – 10 = 0 22. Level 2 (Solving Quadratic Equations)
If x2 + 5x + c = 0 has exactly one solution, what is the value
The quadratic equation above has solutions x = a + b and of c?

x = a – b , where a and b are constants. What is the value a) –2.25
b b) 0
of a ? c) 3.75
d) 6.25

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Polynomial and Quadratic 65 The Ivy

23. 3 26.
2 1 4
3 x 2 − 2 x − = 0 Which of the following quadratic equations has exactly one

Let x = w and x = z be the solutions to the equation above. distinct real solution?
9
What is the value of w + z? a) 16 x 2 – 3x – 4 = 0

a) − 3 b) 9 x 2 – 3x + 4 = 0
2 16
3 16
b) − 4 c) 25 x2 – 25 = 0

c) 0 d) 16 x2 + 25 = 0
3 25
d) 4

24.

2x2 + 5x – k = 0 27.

In the equation above, k is a constant. For what value of k (1 – a) + 3(1 – a)2 = 0

does the equation have exactly one distinct real solution? What are the solutions to the equation above?

a) − 25 a) a = 1
8 4
5 b) a= 3
b) − 4
4
c) 5 c) a= 1 and a = 3
4
25 d) a = 0 and a = − 1
d) 8 3

25. October 2019 28.
x2 – 6x – 11 = 0 3x2 + 2x = 11
If 3 + 2 n is a solution to the given equation, where n is a Which of the following describes the solutions to the
positive constant, what is the value of n? equation above?
a) The equation has two distinct rational solutions.
b) The equation has two distinct irrational solutions.
c) The equation has one distinct real solution.
d) The equation has no real solutions.

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The Ivy 66 Polynomial and Quadratic

29. 31.
If r1 and r2 are the two roots of the equation x2 – 2x – 1 = 0 If (2y + k)2 = 4y2 – 12y + k2, what is the value of k?
and if r1 > r2, what is r1 – r2?
a) 3
a) −2 2 . b) 1
b) –2 c) –2
d) –3
c) 2 2 .
d) 2

30. 32. December 2017
If (px + q)2 = 9x2 + kx + 16, what is the value of k? If (x – 2)2 = 4(x – 2), what is one possible value of x?

a) 3
b) 4
c) 12
d) 24

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Polynomial and Quadratic 67 The Ivy

Quadratic Graph

1. 2.
y = x2 + 4x – 12 y = x2 + 16x + 60

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The Ivy 68 Polynomial and Quadratic
3.
5. March 2019
y = x2 + 12x + 27 y

4. (0, 0) (6, 0) x
y = x2 – 2x – 6 –

–

––

Which of the following is an equation of the parabola shown

in the xy-plane above?

a) y = 1 x(x + 6)
2
1
b) y = 2 x(x – 6)

c) y = –2x(x + 6)

d) y = 2x(x – 6)

6. March 2019
y

2

–O 2 4 6x
–

–

–

Which of the following is an equation of the quadratic
function graphed in the xy-plane above?

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

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Polynomial and Quadratic 69 The Ivy

7. October 2016 10. October 2018 US
In the xy-plane, the graph of the polynomial function f
crosses the x-axis at exactly two points, (a, 0) and f (x) = (x + 3)(x – k)
(b, 0), where a and b are both positive. Which of the
following could define f ? The function f is defined above. If k is a positive integer,

a) f (x) = (x – a)(x – b) which of the following could represent the graph of y = f (x)
b) f (x) = (x + a)(x + b)
c) f (x) = (x – a)(x + b) in the xy-plane?
d) f (x) = x(x – a)(x – b)
a) y
6
4
2

– – –O 2 4 x

8. October 2018 US b) y
y = –(x – 3)2 + a
6
In the equation above, a is a constant. The graph of the 4
equation in the xy-plane is a parabola. Which of the 2
following is true about the parabola? – – –O 2 4 x

a) Its minimum occurs at (–3, a) c) y
b) Its minimum occurs at (3, a) 2
c) Its maximum occurs at (–3, a) 24x
d) Its maximum occurs at (3, a) – – O
–

–4

–6

9. October 2018 US d) y

y 2 24x
k
– – O
–

–4

–6

Ox

The graph of y = 2x2 + 10x + 12 is shown. If the graph
crosses the y-axis at the point (0, k), what is the value of k?

a) 2
b) 6
c) 10
d) 12

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The Ivy 70 Polynomial and Quadratic
13. June 2016
11. December 2018 f (x) = a[(x – a)2 + 1]

y In the function f defined above, a > 1. Which of the
following best represents the graph of f in the xy-plane?
5
a) y
y = f (x) 4
3 (–a, 1) x
2 b) y

1 (–a, a)
x
– – – – – O 12 x
– c) y
–

The graph of y = f (x) is the parabola shown in the
xy-plane. If f (a) = 0, what is the value of a?

a) –9
b) –3
c) 3
d) 9

12. March 2019 US

f (x) = –0.001160(x – 251.5)2 + 73.37 (a, 1) x
d) y
The vertical height, in meters, of the upper arch of the
(a, a) x
Harbor Bridge in Sydney, Australia, above the roadway of
14. October 2016
the bridge can be modeled by the function above, where x is The function f is defined by f (x) = (x + 2)2.
If f (x + a) = x2 – 6x + 9, where a is a constant,
horizontal distance along the roadway, in meters, from the what is the value of a?

entry to the bridge. The graph of y = f (x) is shown in the a) –5
b) –3
xy-plane below. c) 1
d) 3
y Entry to
bridge
Height of arch 100 y = f (x)
(meters) 50
00 100 200 300 400 500 x

Distance from entry to bridge (meters)

In the graph, the point (0, 0) represents the entry to the

bridge. Which of the following points represents the exit

from the bridge on the opposite end?

a) (0, 73.37)

b) (0, 503.0)

c) (73.37, 0)

d) (503.0, 0)

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Polynomial and Quadratic 71 The Ivy

15. May 2019 18. May 2019
In the xy-plane, a parabola has vertex (3, 1) and intersects h(x) = 2(x – 4)2 – 32
the x-axis at two points. If the equation of the parabola is The quadratic function h is defined as shown. In the xy-
written in the form y = –ax2 + bx + c, where a, b, and c are plane, the graph of y = h(x) intersects the x-axis at the points
constants, which of the following could be a value of c? (0, 0) and (t, 0), where t is a constant. What is the value of t?
a) 1
a) –8 b) 2
b) 2 c) 4
c) 3 d) 8
d) 7

16. October 2018 US 19. December 2018
In the xy-plane, a line that has the equation y = c for some The function p is defined by p(h) = 2h + h2. If k > –2 and
constant c intersects a parabola at exactly one point. If the p(k) = p(–2), what is the value of k?
parabola has the equation y = –x2 + 5x, what is the value of
c? a) –1
b) 0
c) 1
d) 2

17. October 2018 US 20. March 2018 US
In the xy-plane, the graph of the function y = x2 – 6x – 16
f (x) = x2 + 5x + 4 has two x-intercepts. The graph of the equation above in the xy-plane is a
What is the distance between the x-intercepts? parabola. Which of the following equivalent forms of the
equation includes the x- and y-coordinates of the vertex as
a) 1 constants?
b) 2 a) y = (x – 3)2 – 25
c) 3 b) y = x(x – 6) – 16
d) 4 c) y = x2 – 2(3x + 8)
d) y + 16 = x(x – 6)

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The Ivy 72 Polynomial and Quadratic

21. October 2019 y = ax2 23. December 2017
y y = bx2 f (x) = x2 – 1
g(x) = –x2 + 3
Ox
The functions f and g are defined above. Which of the
The graphs of y = ax2 and y = bx2, where a and b are following graphs in the xy-plane is the graph of
constants, are shown. Which of the following must be true? y = g(x) – f (x)?

a) 0 < a < b a) y
b) 0 < b < a
c) a < 0 < b 1 x
d) a < b < 0 1

b) y

1 x
1

22. September 2020 y = f (x) c) y
y x

O

1 x
1

The graph of the function f is shown. Which of the following d) y
could define f ?
1
a) f (x) = –x2 – 2x – 3 1
b) f (x) = –x2 + 4x + 3
c) f (x) = x2 – 2x – 3
d) f (x) = x2 + 4x + 3

x

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Polynomial and Quadratic 73 The Ivy

24. May 2018 US 25.
The quadratic function f is defined by f (x) = 2(x + 2)2 – 1. g
In the xy-plane, which of the following could be the graph of
y = f (x) shifted 3 units to the right? 80
60
a) y 40
20
Ox – 0.5 1 1.5 2 2.5 3 3.5 4h

b) y Emma would like to determine how much time she should
Ox spend studying tonight for her math test tomorrow. If she
studies too little, she will not know the material and as a
c) y result get a poor grade. Too much studying will cut into her
Ox sleep and may also prove detrimental. She makes a sketch to
model what she predicts her grade will be, g, based on the
d) y hours she spends studying the night before, h. Assuming that
the graph at left shows a quadratic relationship, which of the
following functions best approximates the relationship
between the hours Emma spends studying and her test grade
for 0 ≤ h ≤ 4?

a) g = –11.5(h + 2)2 + 96
b) g = –11.5(h – 2)2 + 96
c) g = –36.5(h + 2)2 – 96
d) g = –36.5(h – 2)2 – 96

Ox

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The Ivy 74 Polynomial and Quadratic

Quadratic Word Problems 3.
A small business models its weekly revenue (the product of
1. the number of units sold and the price per unit) by the
The surface of a road curves like a parabola to allow water quadratic function
to drain off of it. The following equation shows the surface
height, y, in feet above the base at a point x feet from the left R(x) = 50x – 0.1x2
edge of the road. where x is the number of units sold.
What is the maximum weekly revenue of the business?
y = –0.0015x(x – 40)
The base has the same width as the road. What is the width a) $625
of the road? b) $6,250
c) $25,000
d) $62,500

2. March 2019 US 4.
y = –16t2 + 64t + 80 The "hang time" of a football is the amount of time the
The equation above gives the height of an object above the football stays in the air after being kicked. The height, in
ground, y, in feet, t seconds after it is launched from a meters, of the football above the ground at time t can be
platform. How many seconds after it is launched does the modeled by the quadratic function:
object reach the ground?
h(t)= –4.9t2 + 19.6t
Which of the following equivalent expressions displays the
hang time of the football as a constant or coefficient?

a) –4.9(t – 2)2 + 19.6
b) –4.9(t – 4)t
c) –4.9(t – 3)2 – 9.8t + 44.1
d) –4.9(t – 1)2 – 9.8t + 4.9

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Polynomial and Quadratic 75 The Ivy

5. 7.
Kaia throws a stone vertically upward from a bridge. The The Arc de Triomphe in Paris, France, was completed in the
height, in meters, of the stone above the water at time t can year 1836. The monument has a large open central arch that
be modeled by the quadratic function: can be modeled by the quadratic equation

h(t) = –4.9t2 + 9.8t + 39.2 h(x) = –0.57x2 + 7.98x
After how many seconds does the stone hit the water? where h is the height in meters, and x is the distance from
the left end of the arch at ground level. Approximately how
a) 1 second wide is the arch at ground level?
b) 2 seconds
c) 4 seconds a) 0.57 m
d) 8 seconds b) 7.98 m
c) 14 m
d) 28 m

6. 8. May 2018
Ariel was playing baseball and hit the ball into the air with a
baseball bat. The height, h, in feet, of the ball t seconds after The gas mileage M(s), in miles per gallon, of a car travelling
it left her bat is modeled by the equation below:
s miles per hour is modeled by the function below, where
h(t) = –16t2 + 64t + 4
How many seconds after leaving Ariel's bat does the ball 20 ≤ s ≤ 75. 1
reach its maximum height? 24
M(s) = − s2 + 4s – 50
a) 2 seconds
b) 4 seconds According to the model, at what speed, in miles per hour,
c) 8 seconds
d) 16 seconds does the car obtain its greatest gas mileage?

a) 46

b) 48

c) 50

d) 75

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The Ivy 76 Polynomial and Quadratic

9. October 2017 11.
h(x) = –16x2 + l00x + 10 P(q) = –0.01(q – 250)(q – 80)
The equation above gives the profit, P(q), in dollars, earned
The quadratic function above models the height above the by a cupcake bakery when q cupcakes are produced. What is
ground h, in feet, of a projectile x seconds after it had been the best interpretation of the number 80 in this context?
launched vertically. If y = h(x) is graphed in the xy-plane, a) 80 is a number of cupcakes for which the profit is
which of the following represents the real-life meaning of
the positive x-intercept of the graph? equal to $0.
b) 80 is a number of cupcakes that corresponds to the
a) The initial height of the projectile
b) The maximum height of the projectile maximum profit.
c) The time at which the projectile reaches its c) 80 is a number of cupcakes that corresponds to the

maximum height minimum profit.
d) The time at which the projectile hits the ground d) $80 is the maximum profit, in dollars.

10. October 2018 12.
The expression ax2 + b models the number of rabbits in a h(x) = –5(x – 1)2 + 7
forest, where x is the amount of time passed in weeks since The height, h, of a stream of water, in feet above the surface,
the beginning of the year, and a and b are positive constants. in a fountain at a horizontal distance of x feet from the
Which of the following statements is the best interpretation nozzle, is given by the function shown above. What is the
of b in this context? best interpretation of the number 7 in the function?
a) The stream of water begins at an initial height of 7
a) The number of rabbits in the forest at the
beginning of the year feet above the surface.
b) The stream of water will reach a maximum height
b) The number of rabbits in the forest a weeks after
the beginning of the year of 7 feet above the surface.
c) The stream of water will touch the surface at a
c) The number of weeks since the beginning of the
year when there are at least a rabbits in the forest distance of 7 feet horizontally from the nozzle.
d) The stream of water will reach a maximum height
d) The number of weeks since the beginning of the
year when there are at least a2 rabbits in the forest at a distance of 7 feet horizontally from the nozzle.

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Polynomial and Quadratic 77 The Ivy

13. 15.
A manufacturer of chemical glassware needs to purchase a A dish with 2 grams of nutrient is being used to model the
certain amount of raw material. The profit, p, in dollars change in a microbe population with limited resources. The
expected from purchasing t tons of raw material, where t is growth rate of the population, r, in millions of microbes per
positive, is: hour h hours after introducing the bacteria is given by:

p = 50,000(2t – 1)(t – 5) r = 42h – 3h2
What is the smallest amount of raw material in tons that the How many hours after introduction does the growth rate
manufacturer can purchase to break even with a profit of 0 become negative?
dollars?

14. 16.
A small rocket is fired into the air from the top of a tower.
The height, h, in feet, of a baseball t seconds after Tobin hit Its height h, in feet, after t seconds, is given by:
h(t) = –16t2 + 64t + 32. The rocket is 80 feet above the
it with a baseball bat can be modeled by the equation below: ground twice—once on the way up and then again on the
way down. How many seconds pass between these two
h(t) = –16t2 + 64t + 4 times?

Which of the following equivalent expressions displays the

value of the baseball's maximum height as a constant or

coefficient?

a) –16(t – 2)2 + 68

b) –4t(4t – 16) + 4
1
c) –16(t2 – 4t – 4 )

d) –16(t2 – 4t) + 4

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The Ivy 78 Polynomial and Quadratic

17. 19.
Prachi conducted an experiment to determine the Olivier offers music lessons for $18 per half-hour lesson.
relationship between the number of hours, t, spent studying Students must enroll for a weekly class. Looking at his
the night before the test and the number of questions, q, records for the past five years, he determines that for each
answered correctly on the test. She used the following price increase of $1, he loses about 4 students (and it
equation to model her results. follows, 4 weekly lessons). At the current rate, Olivier has 63
students enrolled. For a change in price of x dollars, which of
q = –3(t – 2)2 + 45 the following functions correctly models Olivier's weekly
What is the best interpretation of the number 2 in Prachi's revenue, in dollars?
model?
a) W(x) = (x + 18)(x – 4)
a) In this experiment, a student who did not study b) W(x) = 0.5(x + 18)(x – 4)
would answer 2 questions correctly. c) W(x) = (18 + x)(63 – 4x)
d) W(x) = 0.5(18 + x)(63 – 4x)
b) In this experiment, a student answered 2 more
questions correctly for every hour of study.

c) In this experiment, a student who studied for 2
hours would answer the most questions correctly.

d) In this experiment, the first 2 hours of studying did
not influence the number of questions the student
answered correctly.

18. March 2019 20.
At a certain college, the economics club plans to sell T-shirts A minor league hockey team has been collecting ticket sales
as a fund-raiser. The president of the club estimates that data over the past year. At a current price of $25 per ticket,
when the price of a T-shirt is $18, there will be 60 T-shirts an average of 4000 seats are purchased. They predict that for
sold, and for every $1 the price of the shirt is reduced, 10 each $1 increase in ticket price, 100 fewer tickets will be
additional T-shirts will be sold. Based on this estimate, what sold. Which of the following functions best models the
is the maximum amount of revenue, in dollars, the club can amount of money that the hockey teams expect to collect
earn from T-shirt sales? (Revenue equals price times number from ticket sales, y, based on an $x increase in ticket price?
sold. Disregard the$ sign when gridding your answer.)
a) y = (25 + x)(4,000 – 100x)
b) y = (25 – x)(4,000 + 100x)
c) y = x(4,000 – 100x)
d) y = 4,000(25 + x)

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Polynomial and Quadratic 79 The Ivy

21. June 2016 Profit (thousands of dollars)23. May 2019
Arlene is a clothing store manager and she set the price for a y
pair of jeans at $40 and is selling 120 pairs each week. She
estimates that if she changes the price by x dollars, where x 1,400
can be positive or negative, she will sell 120 – 3x pairs of
jeans a week, for a total of (40 + x)(120 – 3x) dollars in sales. P(x)
According to her estimate, what will result in the greatest
value of sales? O (25, 0) (55, 0) x
Number of board games
a) Decreasing the price by $2.50 (thousands)
b) Decreasing the price by $1.50
c) Leaving the price unchanged A company produces board games and sells them online and
d) Increasing the price by $2.00 in stores. The quadratic function P models the company’s
monthly profits P(x), in thousands of dollars, when x board
22. games, in thousands, are produced and sold. The graph of
y = P(x), where 25  x  55, is shown in the xy-plane above.
The "break-even point" for a company is the number of units How many board games must the company produce and sell
in order to earn the maximum profit estimated by the model?
sold (other than 0 units) for which:
a) 20,000
b) 40,000
c) 60,000
d) 1,400,000

Profit = Revenue – Cost = 0

Production is profitable only when revenue is greater than

cost. The monthly profit of a company selling x units is

given by the quadratic function:
1
P(x) = − 200 x2 + 30x

Which of the following equivalent expressions displays the

break-even point as a constant or coefficient?
1
a) − 200 ((x – 3,000)2 – 9,000,000)

b) − 1 (x – 3,000)2 + 45,000
200
1
c) − 200 x(x – 4,000) + 10x

d) − 1 x(x – 6,000)
200

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The Ivy 80 Polynomial and Quadratic
24.

An umbrella sprinkler is positioned on a ceiling at a point whose x-coordinate
is 0. Negative values of x indicate distances, in meters, to the left of the
position of the sprinkler, and positive values indicate distances to the right.

The path of water from the sprinkler can be modeled by the quadratic function
1
w(x) = – 4 (x2 – 12)

where w(x) is the height of the water, in meters, at position x.

Which of the following equivalent expressions displays the height of the

ceiling as a constant or coefficient?
1 1
a) – 4 x2 + 3 b) – 4 (x – 12 )(x + 12 )

c) – 1 (x – 2)(x + 2) + 2 d) – 1 (x – 4)(x + 4) – 1
4 4

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The Ivy 81 Exponential Functions

Exponential Expressions and Radicals

1. October 2016 3.

Which of the following is equivalent to a 8 ? 2 3 4 2 3 2
7 What is the value of the expression above?

a) 8 a7 a) 2 3 6
b) 7 a8 b) 4 6 8
c) a 8 a c) 8
d) 16
d) a7 a8

2. December 2016 4.

If x > 0 and x x = xa, what is the value of a? 3 8 y27
3 Which of the following is equivalent to the expression above?
a) 2

b) 3 a) 2y3
4
1 b) 2y9
c) 2 8
c) 3 y 3
1
d) 4 d) 8 y 9
3

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Exponential Functions 82 The Ivy

5. April 2017 8.

Which of the following is an equivalent form of 3 f 6a k2 , 5 98a9 b4

where f > 0 and k > 0? Which of the following expressions is equivalent to the
expression above?
a) f 1 k −1
3a a) 5a3b2 98
b) 10a4b2 7a
b) f 1 k 3 c) 35a4b2 2 a
2a 2 d) 245a3b2 2

c) f 3a k −1

d) f 2 a k 2
3

6. May 2017 US 9.

Which of the following expressions is equivalent to 6 3 640 f 2 g9
Which of the following is equivalent to the expression above?
(16 x 2 ) 1 ?
2 a) 60fg4 3 8 g

a) 4|x| b) 60g3 3 4 f 2
c) 48fg4 3 10 g
b) 8|x|
d) 24g3 3 10 f 2
c) 8 x
d) 16x

7. (Radicals and Rational Exponents) 10. May 2017 US

3 24 v3 w8 Which of the following expressions is equivalent to

Which of the following is equivalent to the expression above? (16 x 9 y 3 ) 1 , where x ≥ 0 and y ≥ 0?
a) 8w5 2
b) 2vw2 3 3
a) 4 x 3 y 3
c) 2vw2 3 3w2 2
d) 8vw2 3 3w2
b) 4 x 9 y 3
2 2

c) 8x3y3

d) 8x 9 y3
2

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The Ivy 83 Exponential Functions

11. 14. April 2017

(6 + 3 )2 If 2 2 x = a, what is 2x in terms of a?
Which of the following is equivalent to the expression above? a
a) 2
a) 15
b) 39 b) a2
c) 39 + 12 3 4
d) 45 + 12 3 a2
c) 2

d) 4a2

12. June 2016 15.

Which of the following is equivalent to ( a+ b ) 2 , where 25h2 +144 h2 + 3 h2
3
Which of the following expressions is equivalent to the

a > 0 and b > 0? expression above?

a) (a + b)3 a) 13h + h 2
3

b) 3 a+ b 1 b) 14h
3
c) ( a+ 2 ab + b ) c) 17h + h 2
3

d) 3 a+ 2 ab+ b d) 18h

13. December 2017 16. (Radicals and Rational Exponents)
If (3 + 5 )(4 – 5 ) = a + b 5 , where a and b are integers,
Which of the following is equivalent to ( x2 − 4 ) 3 ? then what is the value of a + b?
2

a) x3 – 8

b) (x – 2)3

c) ( x2 − 4)3
d) ( 3 x2 − 4 )2

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Exponential Functions 84 The Ivy

17. (Radicals and Rational Exponents) 20. (Level 1) Radicals and Rational Equations
8 ( 2 2 + 8)
16 x2 y16 + 25 x16 y2
Which of the following is equivalent to the expression above? If x and y are positive, which of the following is equivalent to
a) 16 the above expression?
b) 40
c) 8 + 16 2 a) 4xy8 + 5x8y
d) 32 + 8 8 b) 4xy4 + 5x4y

c) xy 16 y14 + 25x14
d) x2 y2 16 y14 + 25x14

18. May 2019 US

If 3 x5 = x a for all positive values of x, what is the value of
x4 b

a ?
b

19. (Structure in expressions) 1
x
Which of the following is equivalent to ( )n ?

a) x n
2

b) x − n
2

c) x n + 1
2

d) x n − 1
2

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The Ivy 85 Exponential Functions

Exponential Equation 4. March 2018
x–4= x+2
1.
If 7 x – 24 = 11, what is the value of x? What value of x satisfies the equation above?

a) 5
b) 7
c) 5
d) 25

2. 5. March 2018
x – 12 = x+ 44
2x – 3 = 14 − 4 x
What is the solution set for the above equation?
a) {5} The equation above is satisfied by which of the following
b) {20}
c) {–5, 20} values of x? 1
d) {5, 20} 2
I. −
3.
t+4 =t–2 II. 0
5
In the equation above, what is the value of the extraneous III. 2
solution, if one exists?
a) I only
a) 0
b) 4 b) II only
c) 5
d) There are no extraneous solutions. c) III only

d) I and III only

6.

8t + 13 = 10 + 4 2t

What value of t is the solution to the above equation?
9
a) 16

b) 9
8
9
c) 4

d) 3
2

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Exponential Functions 86 The Ivy

7. 10.
2h – 5 = h2 + h− 5 3 + 5x = k+6 x

What is the sum of the solutions to the above equation? For what value of the constant k does the above equation have
x = 0 as the only solution?

8. March 2018 11.
Which of the following is a solution to the equation x–5 x+6=0

14 − x + 2 = x? What are the possible values of x that solve the equation
I. –2 above?
II. 1
III. 5 a) 1 and 4
a) I only b) 4 and 9
b) II only c) 9 and 16
c) III only d) 9 and 25
d) I and III

9. April 2019 US 12.
2 9 x – 6 = 10 – 2 x
x−3 = 3 – x
What value of x is the solution to the above equation?
If x is the solution to the equation above, what is the value of

x−3 ?

a) 1

b) 3
2

c) 3
d) 3

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The Ivy 87 Exponential Functions

13. May 2018 16. Level 3 (Radical and Rational Equations)
x+ 28 −2 x+1 = 0
l (l−5) = 6
What value of x satisfies the equation above?
a) 8 What is the sum of all solutions to the above equation?
b) 9
c) 26
d) 27

14. October 2019 17. October 2018 US

(t−2)2 = t – 2 4x =x–3
What are all values of x that satisfy the given equation?
If a is a solution to the equation above, which of the following
must be true? I. 1
II. 9
a) a  2 a) I only
b) a = 2 b) II only
c) –2  a  2 c) I and II
d) a  –2 d) Neither I nor II

15. Level 2 (Radical and Rational Equations) 18.
3 + −6 + 2 x = x
x− x 1 = x 1
What are the solutions to the above equation? 2 2
a) x = –3 and x = 5
b) x = 3 and x = –5 What is the sum of all solutions to the above equation?
c) x = 3 and x = 5
d) x = –3 and x = –5

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Exponential Graph

1. March 2019 2. November 2017 x

y y
30 20
18
25 16
20 14
12
15 10
10 8
6
5 4
x 2

0123 – –– O2 46
Which of the following exponential equations best models the
data shown? The graph of the function f is shown in the xy-plane above.
The equation that describes f (x) = abx, where a and b are
a) y = 2–x integers. What is the value of a?
b) y = 2x
c) y = 3–x
d) y = 3x

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3. June 2016 5. October 2018 US
If y = 5(2x), where x > 0, which of the following is true? y

a) If x increases by 1, then y doubles. 4
b) If y increases by 1, then x doubles.
c) If x increases by 1, then y increases by 5. y = f (x) 2
d) If y increases by 1, then x increases by 5.
– – –O 2 4 6x
4. November 2017 –
United States Consumption of Certain Types
of Renewable Energy since 2003 The graph of y = f (x) is shown in the xy-plane. What is the
y value of f (0)?
655443322055050500 N1um2be3r o4f 5yea6rs7sin8ce9210003x
a) 0
Which of the following equations best models the relationship b) 2
between the variables in the scatterplot? c) 3
d) 4
a) y = –16(60)x
tCoonnnseusmofptiooiln ieqnuimvilallieonntb) y = 16(0.86)x 6. October 2020
c) y = 16(1.14)x
d) y = 16(20)x x012

f (x) 9 12 16

For the exponential function f, the table shows some values

of x and the corresponding values of f (x). The function can

be written in the form f (x) = prx, where p and r are

constants. What is the value of r?

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7. Level 2 (Nonlinear Equation Graphs) 9. March 2019
y
978 y
431526
4
–– –––– 123456789
–– x 2 2 46x
y = g(x)
– – –O
–
–

–

–

An exponential equation is graphed in the xy-plane at left. –
Which of the following equations represents the graph?
The exponential function g is graphed in the xy-plane shown,
a) y = 2(1.75)x where y = g(x). If g(m) = 0, where m is a constant, what is the
b) y = 2(0.75)x value of m?
c) y = 0.75(2)x
d) y = 1.75(2)x a) –8
b) –7
c) 0
d) 3

8. October 2019

y 10. May 2019 US
4 g(x) = 2x – 2
The function g is defined by the equation above. Which of the
y = f (x) 2 following points in the xy-plane is an x-intercept of the graph
of the equation y = g(x)?
– ––O 2x
– a) (–1, g(–1))
b) (0, g(0))
The graph of the function f is shown in the xy-plane. The c) (1, g(1))
function is defined by f (x) = (2)x + k, where k is a constant. d) (2, g(2))
What is the value of k?

a) –4
b) –2
c) 2
d) 4

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11. December 2018 13. April 2019 US
Future Value of an Investment after
y 20 Years for Different Interest Rates

10 $60,000Future value
8 $50,000
$40,000
6 $30,000
$20,000
4 $10,000

2 $00% 5% 10% 15% 20%
Annual interest rate
– – O 24 x
– ccoommppoouunnddeedd adnaniluyally

Which of the following is an equation of the graph shown? An initial investment of $1,000 is made at a constant annual
a) y = 3x interest rate. The graphs above show the corresponding future
b) y = 3x + 1 value v, in dollars, of the investment for different annual
c) y = 2x interest rates, r, after 20 years. One graph shows the value
d) y = 2x + 1 when the interest is compounded daily, and the other graph
shows the value when the interest is compounded annually.
12. March 2019 US Which of the following statements is true?
The graph of the exponential function g in the xy-plane
passes through the points (0, 1), (1, 4), and (2, 16). Which of a) As r increases at a constant rate, v increases more
the following is NOT true? rapidly if interest is compounded annually rather
than daily.
a) A line can be drawn that does not intersect the graph
of g. b) As r increases at a constant rate, v increases more
rapidly if interest is compounded daily rather than
b) A line can be drawn that intersects the graph of g at annually.
exactly one point.
c) As r increases at a constant rate, the difference in
c) A line can be drawn that intersects the graph of g at interest compounded daily and interest
exactly two points. compounded annually increases at a constant rate.

d) A line can be drawn that intersects the graph of g at d) If r = 15% and interest is compounded annually, a
exactly three points. $1,000 investment will be worth $20,000 after 20
years.

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14. Level 3 (Nonlinear Equation Graphs) Number of miles16. December 2016
y
789 y = 2(1.1)x
65
The equation above models the number of miles, y, that Jenna
– – – 3241 1 2 3 4 5 6 7 8 9 x
– plans to run on week x of an 8-week training regimen. Which

An exponential function is graphed in the xy-plane at left. of the following graphs models this situation?

Which of the following equations could represent the graph? a) 8
76
a) y = 2x + 1 5
4
b) y = 2–x + 1 32
1 1
c) y = 2 ( 2 ) x O 12345678Number of miles
Week
d) y = 2( 1 )−x – 1
2 b)
8
15. March 2020 76
5
The function f has the property that for each increase of 1 in Number of miles4
32
the value of x, the value of f (x) is halved. If f (0) = 2, which 1
O 12345678
of the following defines f (x)? Week

a) f (x) = –x + 2 c) 8
1 76
b) f (x) = – x− 2 5
4
c) f (x) = 1 2−x 32
2 1
1 O 12345678
d) f (x) = 2( 2 )x Week

d)Number of miles
8
76
5
4
32
1
O 12345678
Week

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The Ivy 93 Exponential Functions

17. August 2020 18. March 2020
Which of the following is the graph of y = 2x + 1 in the Which of the following is the graph of y = 2x – 1 in the
xy-plane? xy-plane?

a) y a) y
4 4
2
– O 4x
– –O 2 4x b)
– –
– y
4
b) y
4 – O 4x
2 c)
–
– –O 2 4x y
– 4
–
– O 4x
c) y d)
4 –
2 y
4
– –O 2 4x
–
–

d) y
4
2

– –O 2 4x
–
–

–O 4x
–

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19. April 2017

The temperature, in degrees Celsius (°C), of a hot object placed in a room is recorded every five minutes. The temperature of the

object decreases rapidly at first, then decreases more slowly as the object’s temperature approaches the temperature of the room.

Which of the following graphs could represent the temperature of this object over time?

Object temperature (°C)a) 300 b) 300Object temperature (°C)
250 250
200 200
150 150
100 100
50 50
0 0 5 10 15 20 25 30 0 0 5 10 15 20 25 30
Time (minutes) Time (minutes)

Object temperature (°C)c) 250 d) 300Object temperature (°C)
250
200 200
150 150
100 100
50 50
0 0 5 10 15 20 25 30
0 0 5 10 15 20 25 30 Time (minutes)
Time (minutes)

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Exponential Word Problems

1. October 2020 3. October 2020
At the beginning of a study, the number of bacteria in a
A(t) = 50(2)t population is 150,000. The number of bacteria doubles every
hour for a limited period of time. For this period of time,
The function A(t) models the number of liters of a fluid in a which equation models the number of bacteria y in this
population after x hours?
tank after t hours. Which of the following models the
a) y = 150,0002x
number of liters of the fluid in the tank after m minutes? b) y = 150,000(2)x
c) y = x2 + 150,000
a) A(m) = 50(2) m d) y = 2x2 + 150,000
60

b) A(m) = 50(2)60m

c) A(m) = 50(2) 60
m

d) A(m) = 50(2)m

2. October 2020 4. September 2020
Markela opens a bank account that earns interest at a rate of At the start of an experiment, approximately 100 million
2% compounded annually. She puts $200 in the account bacteria were present in a liquid growth solution. Over the
when she opens it and does not make any more deposits into next 6 hours, the number of bacteria approximately doubled
or withdrawals from the account for 2 years. If the amount every hour. Which of the following exponential equation
of money in the account after 2 years is given by the best models the relationship between the number of bacteria
expression 200(1.02)2, which of the following expressions P, in millions, and the amount of time t, in hours, after the
gives the amount of money in the account after 1 year? start of the experiment, where t ≤ 6?

a) 100(1.02) a) P = 100 + 2t
b) 200(1.02) b) P = 2(100)t
c) 100(1.02)2 c) P = 100(2)t
d) 200(1.01)2 d) P = (200)t

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5. September 2020 7.
C(t) = 3.8(1.045)x In the year 1859, 24 rabbits were brought from Europe to
Australia, where they were released into the wild. By 1920,
The cost C(t), in cents, of a postage stamp t years after 1950 there were 10 billion rabbits in Australia. Which of the
can be modeled by the given exponential function, where 0 ≤ following functions, R, could model the rabbit population of
t ≤ 64. What is the best interpretation of the value 3.8 in this Australia t years after 1859?
context?
a) R(t) = 24 + (1.3845)t
a) The model estimates that the cost of a postage b) R(t) = 24(1.3845)t
stamp was 3.8 cents in 1950. c) R(t) = 24(0.6155)t
d) R(t) = 1010(0.97)t
b) The model estimates that the cost of a postage
stamp was 3.8 cents in 2014.

c) The model estimates that the cost of a postage
stamp increased by 3.8 cents each year from 1950
to 2014.

d) The model estimates that the cost of a postage
stamp increased by 3.8% each year from 1950 to
2014.

6. August 2020 8.
A(t) = 55,000(0.79)t Tritium, a radioactive isotope of hydrogen, is often used in
emergency EXIT signs. Because of the radioactive substance
An antibiotic is introduced to a certain culture of bacteria. present in these signs, proper disposal is a matter of concern
After t days, the number of bacteria per millimeter, A(t), to the Environmental Protection Agency. The half-life of
remaining can be modeled by the function A shown. Which tritium is approximately 12 years. That is, every 12years, the
of the following expressions represents the predicted number amount of tritium decreases by 50%. If a new tritium EXIT
of bacteria remaining in each millimeter of the culture x sign contains 25 curies of tritium, approximately how many
hours after the antibiotic is introduced? curies of tritium will remain after 24 years?

a) 55,000(0.79)x a) 6.25 curies
b) 55,000(0.79)x + 24 b) 12.5 curies
c) 25 curies
x d) 30 curies

c) 55,000(0.79) 24
d) 55,000(0.79)24x

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