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Math 2
Math 2 Final Exam – Skills Review
Name __________________________
June 2015
Each of the following is a discrete mathematical skill, followed by practice problems on that
skill. You should be able to complete each of these problems, but you should also be able to
apply each one to solutions of other, more complex, problems.
Chapter 4 – Function Concept
Write an explicit and a recursive function rule for a linear table of values.
1. Fill in the difference column; then write a recursive and an explicit rule for the functions
represented by each of the tables below.
a.
b.
x
0
1
2
3
4
f(x)
9
11
13
15
17
∆
x
0
1
2
3
4
f(x)
3
6
9
12
15
∆
Describe in words and write a recursive function rule for a non-linear table of values.
2. Write a function rule for each table.
a.
b.
x
0
1
2
3
4
f(x)
-2
-1
2
7
14
∆
x
0
f(x)
1
1
2
3
4
3
9
27
81
∆
Describe whether a relation is a function based on an equation, a table, or a graph.
3. Determine whether each rule describes a function or merely a relation. Explain your
reasoning.
a. 𝑦 2 = 2𝑥 − 1
b. 𝑦 = 3𝑥 2 − 2
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4. Determine whether each graph represents a function or merely a relation. Explain your
reasoning.
a.
b.
c.
5. Determine whether each table represents a function or merely a relation. Explain your
reasoning.
a.
b.
x
2
5
2
7
9
y
8
7
6
5
4
x
1
2
3
4
5
y
6
5
6
4
7
Find domain and range of a function based on an equation, a graph, and a table.
6. Find the domain and range of each function. Use interval or inequality notation.
a. f(x) = 3x2 – 6x + 9
b. 𝑓(𝑥) = √𝑥 + 4
2
c. 𝑓(𝑥) = 𝑥+5
7. Find the domain and range of each function shown below. The entire function is shown.
a.
b.
8
4
6
2
4
–5
2
5
–2
–5
5
–4
–2
–6
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8. Determine the domain and range of the function represented by the table below. The table
represents the complete function.
5
7
8
6
x
3
10
12
1
y
Evaluate functions, operations with functions, and compositions of functions.
9. Given the functions 𝑓(𝑥) = 2𝑥 2 + 1 and 𝑔(𝑥) = 4𝑥 − 3, find each value.
a. 𝑓(−2)
b. 𝑔(−2)
c. 𝑓(−2) ∙ 𝑔(−2)
d. 𝑓(𝑔(−2))
e. 𝑔(𝑓(−2))
f. 𝑔(𝑓(4)
10. Given 𝑎(𝑥) = 4𝑥 + 3, 𝑏(𝑥) = 𝑥 2 + 5, and 𝑐(𝑥) = −𝑥 + 6, find a formula for each
composition.
a. 𝑎(𝑏(𝑥))
b. 𝑏(𝑐(𝑥))
c. 𝑎(𝑐(𝑥))
d. 𝑐(𝑎(𝑥))
Find the inverses of functions given as tables, graphs, and equations.
11. Find the inverse of each function.
a. 𝑓(𝑥) = 5𝑥 − 1
b. 𝑔(𝑥) = −3𝑥 + 9
12. Find and draw the inverse of each of the functions. Then determine whether the inverse is a
function.
a.
b.
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Chapter 10 – Coordinate Geometry
Use tools of coordinate geometry to find the midpoint of a segment, find the distance between
two points, find the slope of a segment or line, and determine whether two lines are parallel,
perpendicular, or neither.
13. Quadrilateral ABCD has coordinates A (-12, 0), B (-5, 4), C (3, 2), and D (-4, -2). Find the
length, slope, and midpoint of each side of the quadrilateral.
14. Use the information from the previous question to classify ABCD as a parallelogram,
rhombus, rectangle, square or simply a quadrilateral.
Find the equation of a line based on a description.
15. A circle has an equation (𝑥 + 2)2 + (𝑦 − 4)2 = 100. Find the equations of two vertical
tangents and two horizontal tangents to this circle. Also find the equation of the tangent that
goes through point (–8, –4).
16. Find the equation for a line that is perpendicular to the line 3x – y = 10 and that passes
through the point (-4, 5).
Recognize the basic graphs including: 𝑦 = 𝑥, 𝑦 = 𝑥 2 , 𝑦 = 𝑥 3 , 𝑦 = |𝑥|, 𝑥 2 + 𝑦 2 = 1, 𝑦 =
1
√𝑥, and 𝑦 = 𝑥.
17. Sketch each of the parent graphs below. You should be able to do this from memory.
𝑦=𝑥
𝑦 = 𝑥2
𝑥2 + 𝑦2 = 1
𝑦 = √𝑥
𝑦 = 𝑥3
𝑦=
𝑦 = |𝑥|
1
𝑥
𝑦 = 𝑥3 − 𝑥
Translate basic graphs up down, left and right and determine the resultant equation. Reflect
basic graphs over the x-axis or the y-axis. Stretch or shrink graphs by a given factor. Determine
how any of the preceding transformations changes the equation of the graph.
18. Sketch the graph each of the following equations. Name the parent function and describe in
words how the parent function has been transformed.
a. y -1 = (x - 3)2
1
d. 𝑦 = 2𝑥
b. 2y = -x
e. 2y = x - 5
c. (x -1)2 +(3y)2 = 16
(
)
3
f. -y = x -2 -(x -2)
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19. Each of the following figures is one of the basic graphs with one or more transformation
performed on it. For each one,
i. Identify the parent function.
ii. Describe in writing the transformation(s) that was/were performed.
iii. Write an equation for the new function.
a.
b.
c.
d.
20. Write an equation for each of the following
transformations of 𝑦 = |𝑥|
21. Write an equation for each of the
following transformation of 𝑦 = 𝑥 2 .
a. The graph is shifted left 3 units.
a. The graph is shifted up 3 units.
b. The graph is dilated horizontally by a
factor of 3.
b. The graph is reflected over the yaxis.
c. The graph is reflected over the x-axis.
c. The graph is reflected over the xaxis and shifted left 3 units.
d. The graph is shifted down 5 units.
e. The graph is transformed such that its
range is [−4, ∞).
d. The graph is dilated horizontally by
a factor of ½ (the resulting graph is
½ as wide).
f. The graph is transformed such that the
y-intercept is (0, 3).
e. The graph is transformed such that
its domain is [3, ∞).
f. The graph is shifted left 2 units and
up 3 units.
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Chapter 6 – Deductive Geometry
Identify the pairs of congruent and supplementary angles formed when lines intersect, when
parallel lines are cut by a transversal, and the interior and exterior angles of polygons.
22. Find the measures of missing angles.
A
s
t
v
w
r
q
B
23. Determine whether each statement is sometimes (S), always (A) or never (N) true based on
the figure.
a.
b.
c.
d.
e.
f.
g.
∠8 ≅ ∠9
∠1 ≅ ∠2
∠4 𝑖𝑠 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑡𝑜 ∠5
∠7 ≅ ∠4
∠1 𝑖𝑠 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑡𝑜 ∠8
∠1 𝑖𝑠 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑡𝑜 ∠4
∠7 𝑖𝑠 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑡𝑜 ∠6
8
1
3
9
2
7
6
24. Draw a figure and write a proof for each of the following statements.
a. If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
b. If a point is equidistant from the endpoints of a segment, then it is on the bisector of the
segment.
5
4
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25. For each of the figures below
a) Identify any congruent triangles and write a congruence statement. If there are no
congruent triangles write CBD (for cannot be determined).
b) Write the congruence shortcut used. If there are no congruent triangles, explain why not.
i.
ii.
iv.
iii.
v.
vi.
B
A
C
D
O is the center of the circle.
vii.
viii.
G
G
ix.
H
K
J
D
J
H
F
E
H
I
L
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26. Complete each of the following proofs. Use a separate sheet of paper.
a.
b.
c.
Chapter 1 – Exponents
Simplify radicals.
27. For each equation, find the value of k that satisfies the equation.
a. √98 = 𝑘√2
b. √125 = 𝑘√5
c. √5 + √125 = 𝑘√5
d. 2√6 + √150 = 𝑘√6
28. Simplify as much as possible. Leave no negative exponents. Simplify numerical exponents
when the exponent is 4 or less.
a. ((𝑥 4 )2 )−1
d. (3𝑥 4 𝑦 3 )3
1 5
b. 𝑥12 ∗ (𝑥 3 )
e. (3)5 ∗ (5)5
c. 𝑥 3 ∗ (𝑥 4 )3
f.
6𝑥 7 𝑧 3
2𝑥 3 𝑧
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29. Write each radical in simplified or standard form.
b. √121
a. √32
d.
2
e.
√6
c. √45
5
f.
√3
3
2 √3
Simplify exponents, including fractional exponents.
30. Simplify as much as possible. Leave no negative exponents.
2
3
a. 45 ⋆ 45
1
3
d. 9−2
4
1
3
2
e. (8 )
−
1
c. 276 ⋆ 276
b. (258 )
4
9
Determine whether numbers are rational or irrational.
31. Determine whether each of the following numbers is rational or irrational. Explain your
answer.
a. 16.33
c.
b. √52 − 3
0.125
d. √13
4
Chapters 2 and 3 – Polynomials and Quadratics
Factor Polynomial expressions including
 Finding common factors.
 Difference of perfect squares.
 Monic and non-monic quadratics using trial and error, sums and differences of roots,
splitting the middle term.
32. Factor the following polynomial expressions completely.
a.
b.
c.
d.
e.
f.
g.
h.
i.
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Solve polynomial equations by factoring and using the zero product property.
33. Factor each equation. Then use the zero product property (ZPP) to solve for x.
a.
=0
c.
b.
=0
d.
=0
=0
Use vocabulary to describe polynomials.
34. Write a polynomial that meets each description.
a. Has a degree of 4 and a quadratic term
with a coefficient of 2.
b. A binomial with a degree of 3 and no
linear term.
Add, subtract, and multiply polynomials.
35. Let 𝑔(𝑥) = 3𝑥 3 − 2𝑥 2 + 5.
a. Find ℎ(𝑥) such that 𝑔(𝑥) + ℎ(𝑥) = 4𝑥 3 − 3𝑥
b. Find 𝑗(𝑥) such that ℎ(𝑥) − 𝑗(𝑥) = −𝑥 3 − 2𝑥 2 − 1
c. Find 𝑘(𝑥) such that 𝑔(𝑥) ∗ 𝑘(𝑥) has a degree of 6 and 𝑔(𝑥) + 𝑘(𝑥) has degree 1 and no
quadratic term.
Solve quadratic equations using an efficient method.
Hint: the value of the discriminant of a quadratic equation (𝑏 2 − 4𝑎𝑐) tells you several things
about the solutions.
If the discriminant is…
Then…
Positive
The quadratic has two solutions
Zero
The quadratic has one real solution
Negative
The quadratic has no real solutions
A perfect square
The quadratic is factorable
Not a perfect square
The quadratic is not factorable
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36. Solve the following quadratic equations. Use the most efficient method.
b. −3𝑥 2 + 24𝑥 − 52 = 0
a.
e.
d.
c. 4 = 4(𝑥 − 3)2
f.
Find equations of quadratic functions based on various given information.
37. Write a quadratic equation for each of the following sets of parameters.
a. A parabola with zeros at − 2 and 3 that
passes through (4, 27)
b. A parabola with x-intercepts of – 3 and 4
and a y-intercept of – 24.
c. A parabola with a vertex at (3, 5) passing
through (1, 17).
d. A parabola with a vertex at (2, 7) that
passes through (3, 3).
1
Graph quadratic equations from standard, factored, and vertex forms.
Graph each of the following equations. Clearly label the vertex, the zeros, the axis of symmetry,
the y –intercept, a mirror/sister point:
38. Consider your scale when graphing.
a. 𝑦 = −3(𝑥 − 4)2 + 3
b. 𝑦 = 2(𝑥 − 3)(2𝑥 + 3)
c. 𝑦 = 6𝑥 2 − 13𝑥 − 5
Show the equivalence of multiple forms of solving a quadratic equation.
39. Given the equation 𝑦 = 2𝑥 2 − 9𝑥 − 5
a. Solve by completing the square
b. Solve using the quadratic formula.
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Chapter 5 – Probability
Solve problems about probability experiments.
40. Solve each of the following problems. Show work as well as giving a number.
a. If you flip a coin 4 times, what is the
probability that you will get exactly 2
heads?
b. If you flip a coin 4 times, what is the
probability that you will get at least 2
heads?
c. A spinner has four equal sections labeled 1, 2, 4, and 8. If you spin it twice, what is the
probability that the product of your numbers is 8? (Hint: write out the sample space in
an organized manner.)
d. A bag contains 3 orange, 5 pink, and 2 green marbles. If you draw two marbles in a row
without replacement, what is the probability that both marbles are the same color?
e. Refer to the situation in part d. If you draw three marbles in a row without
replacement, what is the probability that both marbles are the same color?
Use two-way tables to determine independence and conditional probability.
The table below shows polling data about whether students complete homework regularly and
whether they are on honor roll. Use the table to answer the questions that follow.
On Honor Roll (H)
Not on Honor Roll (R)
Total
Always
Completes
Homework
(A)
290
589
879
Sometimes
Completes
Homework (S)
421
229
650
Never
Completes
Homework
(N)
15
656
671
Total
41. Calculate each of the following probabilities and explain in writing what it means.
a. P(A) =
b. P(N) =
c. P(A and N) =
d. P(A or N) =
e. P(H|R) =
f. P(R|N)
g. P(A|H) =
h. P(H|A)
726
1474
2200
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Chapter 8 – Similarity and Trigonometry
Use the properties of similarity to determine lengths and angles between two similar figures.
42. Find the measures of the missing sides and angles.
a.
b.
43. Decide whether the figures are similar. Explain why or why not.
a.
b.
Determine if two triangles are similar using triangle similarity shortcuts.
44. Write a similarity statement (e.g. ∆𝐴𝐵𝐶 ∼ ∆𝐷𝐸𝐹) and a similarity shortcut for each figure.
a.
b.
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45. Prove that ∆𝐷𝐴𝐺 ∼ ∆𝐶𝐴𝑇.
Use the properties of similarity to find the lengths, areas and volumes of similar figures and
solids.
46. Find the value of x in the figure at the right
47. Standing 4 feet from a mirror laying on the flat ground, Palmer, whose eye height is 5 feet, 9
inches, can see the reflection of the top of a tree. He measures the mirror to be 24 feet from
the tree. How tall is the tree?
48. The shadow of a statue is 20 feet long, while the shadow of a student is 4 ft long. If the
student is 6 ft tall, how tall is the statue?
49. Find the missing measures in each pair of similar figures.
a.
b.
Math 2 Final Exam – Skills Review
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Use trigonometric ratios to solve problems.
50. Write a true trigonometric equation and solve for the missing measure. Round to the nearest
tenth.
a.
b.
c.
d.
e.
f.
Draw accurate diagrams to solve word problems using trigonometric ratios.
51. Bharat is flying a kite at the park and realizes that all 500 feet of string are out. Margie
measures the angle of the string with the ground with her clinometer and finds it to be 42.5°.
How high is Bharat’s kite above the ground?
52. Mayfield High School’s flagpole is 15 feet high. Using a clinometer, Tamara measures an
angle of 11.3° to the top of the pole. Tamara is 62 inches tall. How far from the flagpole is
Tamara standing?
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53. Find the missing measures. Give an exact answer.
a.
b.
c.
54. Find the area of each figure.
a.
b. ABCD is a rectangle.
Chapter 8 – Circles
Solve problems using the properties of circles.
55. Find the area of the shaded portion of each figure and the length of each intercepted arc. Give
an exact answer (in terms of pi).
a.
b.
Math 2 Final Exam – Skills Review
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56. Find the missing measure.
a.
b.
57. Find the area of each shaded region or follow the directions in the problem.
a. 𝑚∠𝑀𝐼𝐿 = 90
b.
c.
d. The area of the shaded region is 12π cm2.
Find the radius of the circle.
Math 2 Final Exam – Skills Review
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58. Find the missing measures.
a.
b.
59. Use circle theorems to solve the following problems.
a.
b. Find w, x, and z
D
70°
C
w°
B
x°
z°
Math 2 Final Exam – Skills Review
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Answer Key
1)
9, 𝑥 = 0
a. Recursive Rule 𝑓(𝑥) = {
Explicit Rule 𝑓(𝑥) = 2𝑥 + 9
𝑓(𝑥 − 1) + 2, 𝑥 > 0
3, 𝑥 = 0
b. Recursive Rule 𝑓(𝑥) = {
Explicit Rule 𝑓(𝑥) = 3𝑥 + 3
𝑓(𝑥 − 1) + 3, 𝑥 > 0
2)
a. Explicit Rule 𝑓(𝑥) = 𝑥 2 − 2
1, 𝑥 < 0
b. Recursive Rule 𝑓(𝑥) = {
Explicit Rule 𝑓(𝑥) = 1 ∗ 3𝑥
3𝑓(𝑥 − 1), 𝑥 ≥ 0
3)
a. This is a relation; the same input (x-value) can give you more than one output (y-value).
b. This is a function; each input has a unique output.
4)
a. Function; each x-value has exactly one y-value
b. Relation; for example, the x-value of 1 has two y-values.
c. Function; each x-value has exactly one y-value
5)
a. Relation, the input of 2 has two outputs (6 and 8).
b. Function, each input has only one output.
6)
a. Domain: all real #’s, Range all real #’s y≥ 6
b. Domain: all real #’s 𝑥 ≥ −4, Range all real #’s 𝑦 ≥ 0
c. Domain; all real #’s x≠ −5, Range all real #’s 𝑦 ≠ 0
7)
a. Domain: [2, 3) Range: (3, 6]
b. Domain: (2, 3] Range: [6, 3)
8)
a. Domain: {5, 6, 7, 8} Range {1, 3, 10, 12}
9)
a. 9
b. 11
c. 99
d. 243
e. 33
f. 129
10)
a. 4𝑥 2 + 27
b. 𝑥 2 − 12𝑥 + 41
c. −4𝑥 + 36
d. −4𝑥 + 3
11)
𝑥+1
𝑥−9
a. 𝑓 −1 (𝑥) = 5
b. 𝑔−1 (𝑥) = −3
Math 2 Final Exam – Skills Review
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12)
a.
13)
̅̅̅̅ = 4, mp𝐴𝐵
̅̅̅̅ = (−8.5, 2)
AB = 8.1, m𝐴𝐵
7
̅̅̅̅ = − 1, mp𝐵𝐶
̅̅̅̅ = (−1, 3)
BC = 8.2, m𝐵𝐶
4
4
b.
1
̅̅̅̅ = , mp𝐶𝐷
̅̅̅̅ = (− , 0)
CD = 8.1, 𝑚𝐶𝐷
7
2
1
̅̅̅̅ = − , mp𝐷𝐴
̅̅̅̅ = (−8, −1)
DA = 8.2, 𝑚𝐷𝐴
4
14)
ABCD is a parallelogram, but not a rhombus or a rectangle.
15)
Vertical tangents: x = 12 and x = 8;
Horizontal tangents: y = 14, y = 6
3
Tangent @ (8, –4) 𝑦 + 4 = − 4 (𝑥 + 8)
1
16) 𝑦 − 5 = − 3 (𝑥 + 4)
17) You can easily graph all of these on your calculator, except for 𝑥 2 + 𝑦 2 = 1 which is a circle
with radius 1 centered at the origin.
18)
a. Parent function is 𝑦 = 𝑥 2 , moved right 3
b. Parent function is 𝑦 = √𝑥, reflected over
and up 1.
the y-axis and ½ as tall.
Math 2 Final Exam – Skills Review
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c. Parent function is 𝑥 2 + 𝑦 2 = 16 (a circle
with radius 4). It is shifted right 1 and is 1/3 as
tall.
d. Parent function is 𝑦 = 𝑥 it is ½ as wide.
e. Parent function is 𝑦 = |𝑥|, moved right 5
and ½ as tall.
f. Parent function is 𝑦 = 𝑥 3 − 𝑥, reflected
over the x-axis and moved right 2.
1
19)
a.
i. 𝑦 = √𝑥
ii. moved up 4
2
b.
i. 𝑦 = 𝑥
ii. Right 3, reflected over x-axis
c.
i. 𝑦 = |𝑥|
ii. Left 1, reflected over x-axis
3
d.
i. 𝑦 = 𝑥
ii. Reflect over y-axis (or x-axis)
20)
1
a. 𝑦 = |𝑥 + 3|
b. 𝑦 = |3 𝑥|
c. – 𝑦 = |𝑥|
e. 𝑦 + 4 = |𝑥|
f. 𝑦 − 3 = |𝑥| or 𝑦 = |𝑥 − 3|
21)
a. 𝑦 − 3 = 𝑥 2
b. 𝑦 = (−𝑥)2
c. – 𝑦 = (𝑥 + 3)2
e. Impossible, domain is all ℝf. 𝑦 − 3 = (𝑥 + 2)2
iii. 𝑦 − 3 = √𝑥
iii. – 𝑦 = (𝑥 − 3)2
iii. – 𝑦 = |𝑥 + 1|
iii. 𝑦 = (−𝑥)3 (or – 𝑦 = 𝑥 3 )
d. 𝑦 + 5 = |𝑥|
d. 𝑦 = (2𝑥)2
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22)
a. 124
b. 56
c. 56
d. 38
e. 38
f. 76
g. 66
h. 104 k. 76
n. 86
p. 38
q. 38
r. 76
s. 38
t. 104
v. 76
w. 104
23)
a. Sometimes
b. Always
c. Always
d. Always
e. Always
f. Sometimes
g. Always
25)
i. ∆ 𝑊𝑍𝑉 ≅ ∆𝑌𝑍𝑋 by SAS and ∆𝑊𝑍𝑋 ≅ ∆𝑉𝑍𝑌 by SAS ii. ∆𝑃𝑄𝑅 ≅ ∆𝑇𝑆𝑅 by ASA
iii. CBD (congruent parts are not corresponding)
iv. CBD (not enough parts)
v. ∆𝑀𝑁𝑂 ≅ ∆𝑃𝑁𝑂 by SSS
vi. ∆𝐴𝐵𝐶 ≅ ∆𝐶𝐷𝐴 by AAS
vii. ∆𝐷𝐻𝐸 ≅ ∆𝐷𝐺𝐹 by AAS
viii. ∆𝐺𝐻𝐼 ≅ ∆𝐽𝐻𝐼 by SAS
ix. ∆𝐽𝐾𝐿 ≅ ∆𝐻𝐾𝐿 by SSS
24 & 26 are proofs. Check these with your teacher or compare with a classmate.
27)
a. k = 7
b. k = 5
c. k = 6
d. k = 7
28)
1
1
a. 𝑥 8
b. 𝑥 3
c. 𝑥15
d. 27𝑥12 𝑦 9
e.155
f.3𝑥 4 𝑧 2
29)
√6
3
a. 4√2
b. 11
c. 3√5
d.
a. 4
b. 5
c. 3
d. 27
e.
5√3
f.
3
√3
2
30)
1
1
e. 4
31)
a. rational, can be written as
1633
b. rational, can be written as
100
1
c. rational, can be written as 32
7
1
d. irrational, non-perfect square under radical.
32)
a. (3𝑥 − 5)(2𝑥 + 3)
d. (2𝑥 + 3)(𝑥 + 3)
g. 5𝑥(𝑥 + 4)(𝑥 − 1)
b. (𝑥 − 7)(𝑥 + 6)
e. (𝑥 + 3)(2𝑥 2 + 6𝑥 − 1)
h. (4𝑥 + 1)(4𝑥 − 1)
c. (2𝑥 + 3√2)(2𝑥 − 3√2)
f. (3𝑥 + 4)(3𝑥 − 4)
i. (𝑥 + 8)2
33)
3
3
a. 3𝑥(5𝑥 − 3)(5𝑥 + 3) = 0; 𝑥 = 0, 5 , − 5
b. (5𝑥 − 3)(𝑥 + 3) = 0; 𝑥 =
3
5
, −3
3
c. (𝑥 − 7)2 = 0, 𝑥 = 7
d. (2𝑥 + 3)(𝑥 + 6) = 0; 𝑥 = − 2 , −6
34) Answers will vary, but possible answers are given below.
a. 𝑓(𝑥) = 𝑥 4 + 2𝑥 2
b. 𝑔(𝑥) = 13𝑥 3 − 12
35)
a. ℎ(𝑥) = 𝑥 3 + 2𝑥 2 − 3𝑥 + 5
b. 𝑗(𝑥) = 2𝑥 3 + 4𝑥 2 − 3𝑥 − 4
c. 𝑘(𝑥) = −3𝑥 3 − 2𝑥 2 + 𝑥 +… (any coefficient of x and any constant will work)
36)
4
a. 𝑥 = − 5 , 1
b. no real solutions ∆ = −48
c. 𝑥 = 2, −4
1
d. 𝑥 = 0, −25
e. 𝑥 = − 4 , 3
f. 𝑥 = 0, −4
37)
1
a. 𝑦 = 6 (𝑥 + 2) (𝑥 − 3)
c. 𝑦 = 3(𝑥 − 3)2 + 5
b. 𝑦 = 2(𝑥 + 3)(𝑥 − 4)
d. 𝑦 = −4(𝑥 − 2)2 + 7
Math 2 Final Exam – Skills Review
page 23
38)
a.
b.
Vertex: (4, 3); Zeros (5, 0) and (3, 0); Axis of
symmetry x = 4; y-intercept (0, 45); mirror
point (8, -45)
Vertex (0.75, 20.25); zeros (3, 0) and
3
3
(− 2 , 0); Axis of symmetry x = 4; y-intercept
(0, 18) ; mirror point (1.5, 18)
13
5
1
1
c. Vertex (12 , −12.04); zeros (2 , 0) and (− 3 , 0); Axis of symmetry 1 12; y-intercept (0, −5);
13
mirror point ( 6 , −5)
Math 2 Final Exam – Skills Review
page 24
39)
Completing the square
Quadratic Formula
40)
6
3
a. 16 = 8
11
b. 16
1
c. 4
14
d.45
11
e. 120
41)
879
a. 2200 = 0.4; 40% of students always do their homework.
671
b.
= 0.31; 31% of students never do their homework
2200
c. 0; mutually exclusive, you can’t both always and never do your homework.
d. 0.4+0.31 = 0.71; 71% is the combined percentage of students who do all their
homework and none of their homework
e. 0; mutually exclusive, you can be on honor roll and not on honor roll
656
f. 671 = 0.98; 98% of the students who do no homework are not on honor roll.
290
g. 726 = 0.40; 40% of students on honor roll do all their homework.
290
h. 879 = 0.33; 33% of the students who do all their homework are on honor roll.
Math 2 Final Exam – Skills Review
page 25
42)
a) SL = 5.2
b) AP = 8
MI = 10
EI = 7
m∠𝑈 = 85
YR = 12
m∠𝐷=120
SN = 15
m∠𝐴 = 80
43)
a. Yes, all sides are proportional and all angles are congruent.
b. No, sides are not proportional
44)
a. ∆𝐴𝐵𝐶 ~ ∆𝐸𝐷𝐶 by AA similarity
b. ∆𝑁𝐾𝑂 ~ ∆MKL
45)
∠𝐴𝐺𝐷 ≅ ∠𝐴𝑇𝐶 (Given)
∠ 𝐴 ≅ ∠𝐴 (Reflexive Property)
∆𝐴𝐺𝐷 ~ ∆𝐴𝑇𝐶 by AA similarity
46) x = 33
47) The tree is 34 ft 6 inches tall
48) The statue is 30 feet tall
49)
a. a = 4 cm
b. Ratio of areas = 36:1
50)
a. w = 18 cm b. x = 7.4 cm c. 76.3 cm
51) The kite is 337.80 ft above the ground
52) Tamara is standing 49.2 ft from the flagpole.
53)
a. u = v = 8
b. x = 8√3, y =
12√3
3
d. a = 28.3
e. t = 47.2
c. a = 22, b =11
54)
a. A = 91.18 𝑢2
b. A = 64√3 𝑢2
55)
5
a. Arc Length = 6 𝜋 cm
25
Sector Area = 12 𝜋 cm2
b. Arc Lenth =
16
3
Sector Area =
𝜋 cm2
32
3
𝜋 cm2
56)
a. r = 10 cm
b. x = 135
57)
a. 𝜋 − 2 units2 b. 196 – 49 𝜋2
c.
65
2
𝜋 units2
d. 6 units
58)
a) m∠𝑋𝑁𝑀 = 40
b) a = 50
arc XN = 180
b = 60 c = 70
59)
a. m∠𝐵𝑄𝑋 = 65
b. x = 58
w = 122
c.
z = 58
arc MN = 100
f. z = 76.1