# 10TH GRADE CONTEMPORARY MATHEMATICS IN CONTEXT STUDY GUIDE FOR

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```					   10TH GRADE CONTEMPORARY MATHEMATICS IN CONTEXT STUDY GUIDE FOR
FINAL EXAM

I. CONTENT: Your final exam will have questions based on the following book units and
objectives.

-   BOOK 1 UNIT 7 QUADRATIC FUNCTIONS
o LESSON 1: QUADRATIC PATTERNS
 Recognize quadratic patterns
 Use of tables and graphs to answer questions about quadratic situations.
 Describe effects of parameters on y  ax2  bx  c
o LESSON 2: EQUIVALENT QUADRATIC EXPRESSIONS
 Expand linear factors
 Factor
o LESSON 3: SOLVING QUADRATIC EQUATIONS
 Solve by reasoning, factoring, use of tables and graphs
 Use discriminant and graphs to describe nature of the roots
 Use the quadratic fuormula to solve.

-   BOOK 2 UNIT 1 FUNCTIONS EQUATIONS AND SYSTEMS
o LESSON 1: DIRECT AND INVERSE VARIATION
 Recognize direct and inverse patterns
 Express direct and inverse patterns
 Recognize and represent power functions in graph, equation and table form.
 Solve problems involving direct and inverse variation.
o LESSON 2: MULTIVARIABLE FUNCTIONS
 Write rules of two variables.
 Solve for one variable in terms of another
 Graph linear equations in the form ax + by = c
o LESSON 3: SYSTEMS OF LINEAR EQUATIONS
 Write systems to match given conditions
 Solve linear systems by: graphing, substitution, elimination
 Recognize systems with infinite and no solutions.

-   BOOK 2 UNIT 2: MATRIX METHODS
o LESSON 1: CONSTRUCTING, INTERPRETING, AND OPERATING ON
MATRICES
 Construct and Interpret matrices for real life situations.
 Operate with matrices: Row, Column, Sum, Difference, Scalar multiplication
o LESSON 2: MULTIPLYING MATRICES
 Use matrix multiplication to solve problems.
o LESSON 3: MATRICES AND SYSTEMS OF LINEAR EQUATIONS
 Use matrix equations to solve systems of equations.

-    BOOK 2 UNIT 3: COORDINATE METHODS
o LESSON 1: A COORDINATE MODEL OF A PLANE
   Find distance and midpoints.
   Find slopes of parallel and perpendicular lines.
   Prove properties of quadrilaterals and triangles on the Cartesian plane, using
slopes, distance and midpoint formulas
   Find equations of circles

-   BOOK 2 UNIT 5: NONLINEAR FUNCTIONS AND EQUATIONS
o LESSON 1: QUADRATIC FUNCTIONS AND EQUATIONS
 Function concept and Notation
 Domain and Range
 Factoring and Expanding
 Advanced Factoring
 Solving Quadratic Equations using quadratic formula and factoring
 Break Even Analysis

-   BOOK 2 UNIT 7: TRIGONOMETRIC METHODS
o LESSON 1: TRIGONOMETRIC FUNCTIONS
 Find sine cosine and tangent of angles in degrees and in standard position.
 Find sine, cosine and tangent of right triangles.
 Solve right triangles
o LESSON 2: USING TRIGONOMETRY IN ANY TRIANGLE
 Use Law of Sines and Law of cosines and trigonometric methods to solve
triangulation and indirect measurement problems.

II. Format
Your exam will consist of 3 parts: One multiple choice question section where calculators will not
be allowed, one short answer section where you may use a calculator, and a performance section
where you will have to decide on the appropriate methods to model two real life problems.

III. Materials needed
Please remember to bring to your exam a graphing calculator and your math tool kit. Remember you
will be allowed to use it. If you need a calculator from the resource, please let me know
immediately.

1V. Review Package
Completing the review package will add 5 points to your exam grade.

2011 10TH GRADE REVIEW PACKAGE FOR FINAL EXAM

Multiple Choice
Identify the choice that best completes the statement or answers the question.
____   1. The expression (2x  3)2 is equivalent to
a. 4x2  9 b. 4x2 + 9 c. 4x2  12x  9 d. 4x2  12x + 9 e. 2x2 + 12x + 9
____   2. The solutions to x2 + 5x – 2 = 3 are
a.                 b.                c. x = –2 and x = –4 d. x = –1 and x = –5 e. None of these
____   3. A baseball is hit off a tee that is 1 feet high with an initial upward velocity of 15 feet per second. Which of the
following rules relates the ball’s height above the ground h to its time in the air t?
a. h = –16t2 + 15t + 1 b. h = –15t2 – 25t + 1 c. h = –15t2 – t + 25 d. h = 15t + 1 e. none of these
____   4. Suppose that y is inversely proportional to x with constant of proportionality k = 3.4. What is the value of x
when y = 6?

a. x = -5.4 b. x = 0.4 c. x = 2.5 d. x = 12.6 e. none of these
____   5. The cost of buying s shirts and h hats can be determined using the equation C = 8s + 4h. Suppose that you
have \$200 to spend. Which of the following statements is not true?

a. Each shirt costs \$8. b. Each hat costs \$4. c. Eight shirts and 12 hats will cost \$112. d. You can buy 15
shirts and 18 hats. e. You can buy 20 shirts and 15 hats.

____   6. The graph of the function y =         is shown below. Which of the following must be true about k and n?

a. k 0 and n is even. b. k 0 and n is odd. c. k 0 and n is even. d. k         0 and n is odd. e. It is not
possible to say anything about the values of k and n.
____   7. The number of children at the Boys and Girls Club each day last week is given by the matrix N.

Next week is a school vacation week, and the Club is expecting a 30% increase in the number of children who
attend each day. Which of the following will give you a matrix indicating the number of students expected at
the Club each day next week?

a. 1.30N b. 1.30 + N c.          d. 0.30N e. 0.30 + N
____   8. If A is a 3   4 matrix and B is a 2    3 matrix, then the size of B   A is:

a. 6   12 b. 3   3 c. 3    2 d. 4     2 e. 4   3
____    9. Let A =                 and B =              . Find the value of x so that the second row second column entry in A   B

is 5.

a. x = -9 b. x = -2 c. x = -1 d. x = 0 e. x = 1.4
____ 10. A circle in the standard (x, y) coordinate plane has center at (-5, 5) and contains the point (-5, 0). Which of the
following is an equation for the circle?

a.                     b.                  c.          +            = 25 d.           +              = 5 e.    +

= 25

____ 11. What is the slope of any line parallel to the line 6x - 2y = 12?

a.      b.        c.        d.   e. 3

____ 12. What value of k will make the line kx + 6y = 10 perpendicular to the line y =                    + 4?

a. k = -9 b. k = -4 c. k =          d. k = 4 e. k = 9

____ 13. What is the distance between the points (2, -3) and (-1, 5)?

a.           b.         c. 10 d.            e. none of these
____ 14. The graph of a quadratic function opens down and has a maximum point of (2, 7). Which of the following
could be the x-intercepts of the graph?

I. (1, 0), and (4, 0)           II. (-8, 0) and (12, 0)      III. (0, 0) and (4, 0)

a. I only b. II only c. III only d. II and III e. It is not possible to determine anything about the x-
intercepts.
____ 15. Which of the following equations has exactly one solution?

a. x - 1 =        b.    -          c.      +2 -6=          - 8 d.    =     - 25 e. - + 3 =

____ 16. To the nearest degree, what is the degree measure of the angle formed by the line with equation                    and
the positive x-axis?

a. 37° b. 39° c. 51° d. 53° e. 144°

____ 17. In right triangle ABC, the cosine of               is . Which of the following is sin        ?
a.    b.        c.      d.   e.

18. h( x)  2 x 3  x  2 Find h(-1)
A) 0                   B) -1                     C) 1                    D) 5

19. Identify the property(ies) that justify equivalence of the following pair of algebraic
expressions:
(a + 2b) +3c = a + (3c + 2b)

I. Associative (+)                II. Distributive (x, +)         III. Commutative (+)

(A) I only             (B) II only             (C) I and II only               (D) I and III only

20. Suppose h(x) is a quadratic function with the zeros indicated below. Which of these
is the standard polynomial form of this function: h(-2)=0 and h(5)=0

( A)h( x)  x 2  3x  10 ( B)h( x)  x 2  3x  10 (C )h( x)  x 2  3x  5 ( D)h( x)  x 2  3x  5

21. Which of the following graphs is not a graph of a function?

a.                                b.                                      c.
d.

22. In a right triangle ABC with right angle C and sides a, b, c, which of the following
represents sin A
B
c            a

A        b    C

a                        a                   b                         b
A)                       B)                  C)                        D)
b                        c                   c                         a

23. In the same right triangle of question 10, if you knew the measure of angle B, and side
a, which trigonometric function would be most appropriate to use to find b?

(A) sin B                    B) sin A            C) cos A                  D) tan B

24. For a right triangle ABC, with right angle C which of the following statements must
be incorrect?
A) sin A  3               B) tan A  4         C) tan A  0.5            D)
4
4
cos A 
3

Short Answer

1. Consider the graph of the equation y = 2x2 – 6x.

a. Without using your calculator, find the x-intercepts of the graph.

b. Without using your calculator, find the minimum or maximum point of this graph.
2. A height of a softball, in feet, that has been pitched by a slow-pitch softball pitching machine is given by the
rule h = –16t2 + 30t + 2.5 for any time t seconds after it is pitched.

a. Explain the meaning of the –16, the 30, and the 2.5 in the equation.

b. How long is the ball in the air?

c. What is the maximum height that the ball reaches? When does it reach that height?

d. At what time(s) is the ball at least 10 feet above the ground?
3. Chris can buy ice-cream bars for 20¢ each. Based upon experience she knows that the function rule n = 150 –
100p will give a good estimate of the number of ice-cream bars she will sell in one day if she charges p
dollars for one bar.

a. Write a function rule that will give the income Chris can expect if she charges p dollars for each ice-cream
bar.

b. Write a function rule that will give the profit that Chris will make each day if she charges p dollars for one
ice-cream bar.
c. For what price(s) will Chris make at least \$15 per day?

d. What is the maximum amount of profit that Chris can make, and what price should she charge to make the
maximum profit?

e. How many ice-cream bars will Chris sell if she makes the maximum profit?
4. Rewrite each of the following in standard quadratic form.

a. 2x(4x – 15)

b. 3x(2x + 1) - 3(2x + 1)

c. (x - 4)(x – 3)

d. (x + 3)(x – 3)
5. Solve each equation by reasoning with the symbols themselves.

a. 2x2 + 10 = 17

b. 8x2 + 5x = 0

c. x2 – 7x + 12 = 0

d. 3x2 + 4x + 1 = 0

e. 5 = x2 + 6x

6. To answer the following, refer to the equation z =     where x, y, and z are all positive.
a. If x is held constant and y increases, how does z change?

b. If y is held constant and x increases, how does z change?

c. Write an equivalent rule that shows x as a function of y and z.

d. Write an equivalent rule that shows y as a function of x and z.
7. The time required to complete a 100-mile bike race is inversely proportional to the average speed that the
rider maintains.

a. Write a rule that expresses the relationship between average speed s and race time t.

b. What is the constant of proportionality for this situation?

c. Tina took 5 hours and 15 minutes to complete the race. What was her average speed?

d. Gregory maintained an average speed of 16 miles per hour. How long did it take him to complete the race?
8. Towne Sporting Goods establishes a selling price S for an item based on the cost C that it paid the
manufacturer and the rate R of markup that it charges in order to cover its expenses and make a profit. These
variables are related by the following equation:
S = C(1 + R)

a. Towne Sporting Goods gets a pair of in-line skates from the manufacturer at a cost of \$80. If Towne uses a
28% markup, what is the selling price of the skates to the nearest dollar?

b. Use the equation S = C(1 + R) to write an equivalent equation that gives C as a function of S and R.

c. After the holidays, Towne Sporting Goods had a sale during which it sold all items in the store for a
markup of only 10%. The sale price of a tennis racket was \$32. To the nearest dollar, how much did it cost
Towne Sporting Goods to buy the racket from the manufacturer?
9. Draw a graph of the equation 5x - 3y = 24.
10. Consider the following system of equations:

y = 2x – 10
3x + 4y = 15

a. Use an algebraic method to solve this system of equations. Show your work.

b. How does the solution you found in Part a relate to the graphs of the two equations?
11. Joe looked at the following system of equations and announced that the system had infinitely many solutions.

6x + 8y = 24
9x + 12y = 24

Is Joe correct? Describe how you can determine this just by looking at the equations.
12. A system of linear equations can have 0, 1, or infinitely many solutions.

a. Write a system of equations that has no solution. Explain how you know the system does not have a
solution.

b. Write a system of equations that has exactly one solution. Explain how you know it has exactly one
solution.

c. Write a system of equations that has an infinite number of solutions. Explain how you know there are an
infinite number of solutions.
13. Determine whether each statement is True or False. It False, explain your reasoning.

a. You can add any two matrices together.

b. For all matrices A and B that can be added together, A + B = B + A.

c. For all matrices A and B that can be multiplied together, A   B=B     A.

d. If A =         , then                  .
e. For any 2   2 matrix A, it is always possible to find a matrix B such that A   B = I where I is the identity
matrix.

f. When you add two matrices together, you add the corresponding entries.
14. The Burlington-Edison School District athletic director is writing a funding proposal to the school board for
new athletic equipment for the five middle schools (I–V) in the district. She decides that the schools need a
variety of equipment including baseball bats (BB), volleyballs (VB), and football helmets (FH). The matrix,
Equipment Request by School, summarizes her request.

a. The equipment can be ordered at different levels of quality and cost. The matrix below gives the cost of
each piece in dollars.

Use a matrix operation to complete the total-equipment-cost matrix below. Show how you obtained this
answer.

b. If school V orders average-quality equipment instead of high-quality equipment, how much will the school
save? Show your work.

c. Find the total savings if the athletic director orders average-quality equipment for all schools.

d. The athletic director decides to order high-quality football helmets for safety reasons but only average-
quality baseball bats and volleyballs.

•       Construct a 1 3 cost matrix that gives the cost of each piece of equipment that is requested.
Be sure to properly label your matrix.

•       Then find a matrix for total cost per school. Indicate the matrices to be multiplied. Be sure to
properly label your total cost per school matrix.
15. Refer to the following system of linear equations.
3x + 2y = 15
x-y=2

a. Graph this system of equations and estimate the (x, y) pair that solves it.

b. Check your estimate in Part a. Was your estimate correct?

c. Write a matrix equation that represents this system of linear equations.

d. Solve the system of equations using matrices. Show or explain your work.
16. Pablo is remodeling parts of his house. The bill for work done in his study was \$146.10 and covered 3 sheets
of paneling and 2.25 hours of labor. The bill for work done in his basement was \$457.10 and covered 8 sheets
of the same paneling and 7.5 hours of labor. Use a system of equations to determine the hourly charge for
labor and the cost of each sheet of paneling.
17. Is the quadrilateral ABCD with vertices A(-3, -2), B(-1, 2), C(4, 3), and D(3, -1) a parallelogram? Provide a
mathematical argument that supports your answer.
18. Quadrilateral ABCD is a rhombus.

a. Determine the coordinates of point C. Show your work and explain your reasoning.

b. Prove that           .

c. Prove that     bisects    .
19. Shown below is a circle with radius 4 and center at the origin. Identify the coordinates of two points that are
on the circle and are not on the x- or y-axis. Show your work or explain your reasoning.

20. The graph below shows the height (in meters) of a baseball in flight as time (in seconds) passes and       y=
h(x).

a. Why is it correct to say that height of a baseball is a function of time in flight?

b. Is time in flight of a baseball a function of the height of the baseball? Explain your reasoning.

c. What does the equation h(1.2) = 16.8 tell about the flight of the ball?

d. What is the value of h(3) and what does it tell about the flight of the ball?

e. Estimate the values of x that satisfy the equation 10 = h(x), and what do those values tell about the flight of
the ball?

f. Identify the practical domain and range of h(x).

g. The maximum value of the graph is 20 and the x-intercepts are (0, 0) and (4, 0). Find a function rule for
h(x).

h. Identify the theoretical domain and range of h(x).

21. Consider the equation        -7 +6=0
a. Solve the equation algebraically.

b. Explain how you could solve the equation using a different algebraic method.

c. Solve the equation using technology.
22. The graph of a particular quadratic function has one of its x-intercepts at (4, 0) and a minimum point of (0, -
16).

a. What is the other x-intercept of the function? Explain your reasoning.

b. Find a function rule for this quadratic function.

c. If possible, find a function rule for another quadratic function that has the same x-intercepts as this function
but has a different y-intercept. If not possible, explain why not.

23. Write each product in equivalent                form.

a. (x + 6)(x - 12)

b. (x - 7)(x + 7)

c. (x + 8)

d. (3x - 7)(4x - 6)
24. Write each quadratic expression in equivalent factored form.

a. x + 2x - 35

b. x - 8x + 16

c. 2x + 16x + 30

d. 3x - 22x + 7
25. Use algebraic reasoning to solve each equation. Show your work.

a. x + 5x + 15 = 3

b. x - 36 = 0

c. 3x + 5 =      - 6x + 9

d. x + 4 =

26. The tractor-pulling contest is one of the most popular contests each year at the Johnson County Fair. Based on
data from previous years, the organizers can expect that income I(p) and expenses E(p) both depend on the
price of admission. They predicted that
-6   + 100x and E(p) = 15x + 230.

a. Use algebraic reasoning to determine the ticket price(s) for which income will equal expenses.

b. Write a rule that gives predicted profit F(p) as a function of the admission price.

c. Use your profit function to determine the maximum predicted profit.

d. What admission price should they charge in order to get the maximum predicted profit?
27. The Great Pyramid of Cheops in Egypt has a square base 230 meters on each side. Each face of the pyramid
makes an angle of approximately 52 with the ground. A sketch of the pyramid is shown below.

a. Find the height of the pyramid.

b. Find the area of one triangular face of the pyramid.
28. The pitcher’s mound on a softball field is 40 feet from home plate, and the distance between bases is 60 feet.
As shown in the diagram below, m           = 45°. (Note:        is not a
right angle.)

a. How far is the pitcher’s mound P from first base F?

b. Find m        .
29. Find the indicated measures for each triangle.

a.
AB = ___________

AC = ___________

b.

___________

m     ___________

MK = ___________

30. Given right triangle ABC. If C is the right angle, AC = 5 3 and BC = 5 what is the
measurement of AB, and the measurements of angles A and B?

AB=

m<A=___________
m<B=___________

Factor
31. 25x 2 - 64y 4
32. 2 xy  6 y  xz  3z

Rewrite the following expressions in standard polynomial form.

33. ( (2 x 3  4 x)  3(4 x 3  x 2  5 x  9)

34. g ( x)  2(6  3x)(4 x  x)
4 x  x)

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