Honors Math 2
Name:
Date:
Final Exam Review – Geometry Problems
x2+10
1. Find values of x that would make m // n.
m
3x+17
n
D
C
2. Given: Circle O with DC // AB
Prove: AD @ BC (Hint: Draw OD and OC )
D
B
O
3. AB // DC and AD // GC.
Solve for x and y.
(Hint: Look for ~ )
9
7
x
A
8
F
B
4
y
4. Given: AD @ BC
Prove: AB is not @ CD
A
C
B
C
G
(Hint: Draw AC)
P
K
D
5. Given:
kite KITE
Find the 3 possible values of the perimeter of KITE
A
5y+6
I
E
AB @ AC
AE @ DE @ DB @ BC
Find measure of ÐA (not drawn to scale)
6. Given:
B
y2
2y+3
E
T
A
D
7. Given:
PR = PS
RV bisects ÐPRS
SV bisects ÐPST
1
mÐV = mÐP
2
(Hint: let mÐP = 4x )
C
P
V
W
Prove:
R
S
T
8a. Find a formula for the total number of diagonals in any n-gon.
b. Show that no polygon exists in which the ratio of the number of diagonals to the sum of the
measures of the polygon’s angles is 1 to 18.
9.
E
Given: GB^AC
HD^EC
JF^AE
AB = 8, BC = 12, EC = 15,
AE = 10, GH = 5
Find: GJ and HJ
(Hint: Look for ~
10.
D
F
G
J
H
A
)
C
B
G
4
Given: FG^GH
Ð1 is complementary to Ð3
Prove: GH 2 = JH× HF
3
2
H
1
J
F
11. The medians of a right triangle that are drawn from the vertices of the acute angles have lengths of
2 13 and 73 . Find the length of the hypotenuse.
12. Quadrilateral QUAD has vertices at Q(-7,1), U(1,16), A(9,10), and D(1,-5).
a. What type of quadrilateral is QUAD? Justify your answer.
b. Find the perimeter of QUAD
Q
13. In circle O, PQ = 4, RQ = 10, and PO = 15.
Find PS (the distance from P to circle O).
P
R
S
O
14. A circle is inscribed in a triangle with sides 8, 10, and 12.
The point of tangency of the 8 unit side divides that side
in the ratio x:y, where x < y. Find the ratio.
15. Given trapezoid WXYZ, find the ratio of the areas of each pair of triangles.
a.
and
b.
and
c.
and
d.
and
e.
and
Z
12
W
X
P
16
16. Write a coordinate proof of the following: “Given a square, if the midpoints of the four sides are
joined to form a new quadrilateral, prove that this new quadrilateral is also a square.” Include a
diagram that illustrates your proof.
17. Given an isosceles triangle with vertices (0, 0), (a, b), and (2a, 0), what are the coordinates of the
triangle’s circumcenter (intersection of the perpendicular bisectors of the sides)? Show work
supporting your answer. You may assume that all pertinent theorems have already been proved.
Y
18. Given Circle O, and the following arcs, find
the measure of each numbered angle.
E
D
8
arc AG = 72°, arc GF = 74°, arc DE = 50°,
arc CD = 52°, arc BC = 46°
F
5
C
O
9
6
10
7
1
G
4
B
A
3
2
H
Final Exam Review – Algebra Problems
Function Concept (Chapter 4A, 4B, 4D)
1. a. Which of the following recursive functions best fits the table below?
Input, n
Output, a(n)
0
2
1
-1
2
-7
3
-19
4
-43
5
ì
2
if n = 0
i. a(n) = í
î a(n -1) - 3 if n > 0
ì
2
if n = 0
ii. a(n) = í
î2× a(n -1) - 5 if n > 0
ì
2
iii. a(n) = í
î a(n -1) + 2n - 5
ì
2
iv. a(n) = í
î a(n -1) - 2n - 2
if n = 0
if n > 0
if n = 0
if n > 0
b. Fill in the value for a(5) in the table using the definition you chose.
2. a. Complete the difference table for the
quadratic function f(x).
b. Find a closed-form function that
agrees with the table.
x
f(x)
0
3
3
1
2
c. Find a recursive function that agrees
with the table.
7
2
3
4
5
3. For each function, find the domain and range.
a. f (x) = 3x 2 - 6x + 9
2
c. f (x) =
x+4
b.
32
f (x) = x + 4 - 2
4. Consider the functions f (x) = 2x 2 +1 and g(x) = 4x - 3. Find each value or formula.
a. g f (-2)
b.
f (a +1) - f (a)
c. f  g(x)
d.
g f (x)
5. Below are the graphs of three functions. Which functions are one-to-one, if any?
6. The graph shows a piecewise-defined
function.
a. Write a function for this graph.
b. Write the piecewise definition for the
inverse of this function.
c. Carefully sketch the graph of the
inverse of this function.
7. Find the formula for f
f (x) =
x -4
x+3
-1
2
2
(x) and state the domain of the function and its inverse.
8. Sketch each of the basic graphs below. You should be able to do this from memory.
y=x
y = x2
y = x3
y= x
y = 1x
x 2 + y 2 =1
y= x
9. Sketch each of the transformations of the basic graphs below. Describe how the basic graph
has been transformed.
c. y = - x - 5
d. y = 2x1
a. y - 2 = (x + 4)2
b. 2y = -x
10. Let f (x) = x 4 + x 2.
a. Show that f is an even function by showing that f (-x) = f (x).
b. Show that f is an even function by using the graph of f.
11. Let f (x) = x 4 + x .
a. Find the x-intercepts of the graph of f.
b. Find the x-intercepts of the graph of g(x) = (x - 2)4 + (x - 2) .
c. Sketch the graph of g(x) = (x - 2)4 + (x - 2) . How are the graphs of f and g related?
Exponents and Radicals (Chapter 1)
12. Here is an input-output table for the function g(x) = x 5.
Simplify each square root.
32
1024
7776
a.
c.
e.
243
3125
b.
d.
f.
What is the pattern?
13. Draw a diagram of the sets of Z, Q, and R. Then place each number in the
diagram.
2
28 37
15 +
+
5 -3
3× 7
a.
b.
c.
d.
1000
3
9
3
× 55
16 + 25 g.
15 + 9
e.
f.
h.
4.012
11
14. Decide whether each expression equals 34 . Explain.
1
15. a. What number is defined to be 16 4 ?
-1
b. What is 16 4 ? Use a law of exponents.
1
16. Show that 27 6 = 3 .
16A. Write each expression as a single power of x. Assume 𝑥 ≠ 0.
a. 𝑥 −2 ∙ 𝑥 −1
𝑥4
d. 𝑥 −4
1 −2
b. (𝑥)
c. ((𝑥 3 )−2 )−5
e. (𝑥 6 )0 ∙ 𝑥 3
f.
(𝑥 5 )(𝑥 −2 )
𝑥 10
16B. Simplify each expression without a calculator.
1
3
a. 49−2
4
d. ( √16)
4
c. 8−3
b. 814
5
e. 64
−
2
3
7
f.
175
2
175
Polynomial and Quadratic Functions (Chapters 2, 3A, 3B)
17. a. What values of m make 64t 2 + mt + 9 a perfect square trinomial?
b. What values of n make 81t 2 + 90t + n a perfect square trinomial?
18. Find a value of p such that 2x 2 - 8x + p has the following solutions.
a. two real-number solution
b. one real-number solution
c. no real-number solutions
19. Find a quadratic equation for the given roots.
a. 12 and 4
b. 3 + 5 and 3 - 5
20. Solve the following quadratic equations.
a. x 5 = x 3
b.
d. (3x - 5)(x + 2) = -8
e.
g. 8(x +1)2 - 2(x +1) - 3 = 0
h.
Use the most efficient method.
4(x - 3)2 = 4
c.
5x 2 - x - 4 = 0
f.
x 2 -10x = 20
16x 2 - 32x - 9 = 0
i.
14 + 9x = 8x 2
4x 2 +12x = 9
21. Sketch the graph of y - 4 = -3(x + 5) 2 and find the vertex and line of symmetry.
22. A quadratic function has zeros 3 and 9. It passes through point (6,11). What is the vertex
of the graph?
Probability
1. Consider flipping a coin eight times.
a. Find the probability you will get exactly five heads and three tails.
b. Find the probability you will get exactly four heads and four tails.
2. In a game, you roll a standard number cube and flip a coin. The coin has the number 2 on one
side, and 6 on the other side. Your score is the sum of the values that appear of the number
cube and the coin flip.
a. You win the game if you score 8 points or more. Find the probability that you win.
b. You get to roll the number cube and flip the coin a second time if you score 5. Find
the probability that you score 5.
c. Are the scores from 3 to 12 equally likely? Explain.
3. You draw two cards out of a standard deck of cards.
a. What is the probability of drawing an even numbered card, then drawing a 10 with
replacement?
b. What is the probability of drawing an even numbered card, then drawing a 10 without
replacement? (Hint: there are two cases to consider here)
4. An urn contains 7 blue marbles, 5 red marbles, and 2 white marbles. If you draw two
marbles, what is the probability that they are different colors?
Volumes
5. A pyramid has a height of 5 cm. Its base is a rhombus with
diagonals measuring 7 cm and 6 cm. Find the volume of the pyramid.
6. A hole with a diameter of 2 inches is drilled through a block as shown.
Find the volume of the resulting solid to the nearest cubic inch.
7.
Find the volume of the prism shown at the
right.
8. An ice-cream cone is 9 cm deep and 4 cm across the top. A single (spherical) scoop of ice
cream 4 cm in diameter, is placed on top.
If the ice cream melts into the cone, will it overflow?
(Assume the ice cream's volume does not change as
it melts.) Justify your answer.
Similarity
9.
Given: Parallelogram YSTW
X
Y
W
V
SX^YW
SV^WT
Prove: SX× YW = SV× WT
T
S
C
10.
Given: Parallelogram ABDF
Prove:
B
D
E
A
F
C
D
Given: DH || BC
11.
HF || BG
Prove:
CD
DE
B
E
H
= GF
FE
F
G
Trigonometry
12. Suppose that you are on a salvage ship in the Gulf of Mexico. Your sonar system has located
a sunken Spanish galleon at a slant distance of 683 m from your ship, with an angle of
depression of 28 .
a. How deep is the water at the location of the galleon?
b. How far must you sail to be directly above the galleon?
c. You sail directly toward the spot over the galleon. When you have gone 520 m, what
should the angle of depression () be?
13. In the following problem, you will derive the exact value of cos36°. In isosceles
the angle bisector of angle B has been drawn in. Let this bisector BD be called x.
a.
by AA. Write a proportion
involving AB, BC, and CD and solve for x.
b. Divide
into two right triangles and
solve for the exact value of cos36°.
,