FINAL EXAM REVIEW Name______
ALGEBRA ONE JUNE 2017 level 3 block______date______
Most of the final exam will focus on second semester. First semester topics are limited to the basics which are necessary for any future math course that you take.
This study guide gives a sample of each type of problem, but it should not be the only source of study problems for you. Refer back to the study guides you have done for each of the above chapters and go over any quizzes and TESTS you have taken this year. There will be review days in class – but you have to have specific questions. Check answers in your work with the answer key posted on my website. Any work done on a separate piece of paper must be neatly and clearly numbered.
Must show work on applicable problems to get credit. This study guide is worth 34 participation points (2 points per page – NOTE that work must be shown to get credit).
THIS COMPLETED PACKET IS DUE ON THE DAY OF OUR EXAM (Tuesday June 20) AT 8 am – NO EXCEPTIONS
First Semester Topics
Equation Solving
Be able to solve two step, multi-step, variables on both sides, those containing fractions & those with special solutions. Remember to show all work, each step on a separate line & whatever you do to one side of the equation you do to the other. It’s a good idea to CHECK YOUR SOLUTIONS! SHOW WORK
1) 2) -12 – x = 4 3)
4) 7x – 3x – 6 = 24 5) 5x +3(x + 4) = 28 6) 4x – 3(x – 2) = 21
7) 8) 9)
10) 3x + 6 = 10 – 3(x – 2) 11) 4(3x – 5) = 2(x – 8) – 6x 12) 5(x – 4) = 5x + 12
13) 3(x + 5) = 3x + 15 14) 15) 24w – 8 – 10w = -2(4 – 7w)
Graphing Linear Functions:
Graph the following equations the method appropriate for the form. Label intercepts when possible with ordered pairs on the graph.
16) y = x + 3 17) y = -2x 18) y = -x – 5
19) y = 4 20) y = x 21) x = – 5
22) y + 4 = 2(x – 5) 23) 24) 2x + 3y = 6
25) x + y = -3 26) 4x – 2y = 12 27) y – 3 = -(x + 4)
UNIT 6 – SYSTEMS OF LINEAR EQUATIONS & INEQUALITIES
Solving Systems by Graphing-
28) Solve the following system by graphing each
Line and finding the intersection point.
2x + y = 6
y = x – 3
Solving Systems by Substitution- remember you must get one variable alone first and then plug it into the other equation. Then solve and substitute your result into either original equation to find the other variable. Check your solutions in both original equations. Use separate paper if necessary.
29) 3x + y = 3 30) x = -y + 1
7x + 2y = 1 2x – y = 8
Solving Systems by Linear Combinations- remember you want one of the variables to cancel out when you add the equations together. You may need to multiply one or both equations by some number to get a pair of opposites that will cancel. Remember to solve for both variables and check your solutions in both original equations. Use separate paper if necessary.
31) 5x + 2y = -10 32) 2x – 3y = -7
-4x + 3y = 8 3x + y = -5
Special Types of Systems – remember there are two special solutions “no solution” and “infinite solutions” and they occur when both variables cancel. Know which is which! No partial credit on this.
33) -7x + 7y = 7 34) 4x + 4y = -8
2x – 2y = -18 2x + 2y = -4
Applications of Systems of Equations – remember to identify your unknowns, write a system of equations for the conditions, solve the system and answer the question!
35) A group of 40 children attended a baseball game on a field trip. Each child received either a hot dog or bag of popcorn. Hot dogs were $2.25 and popcorn was $1.75. If the total bill was $83.50, how many hot dogs were bought? How many bags of popcorn were bought?
36) The local pet store has a surplus of cats and canaries. The inventory shows there are 16 heads and 50 legs. How many of each animal are there?
38) You have $50,000 to invest on the stock market. You plan to invest part of it in a high risk venture that has a return of 8% and the rest in a low risk venture that has a more modest return of 2%. If your total return the first year was 3.8%, how much did you invest in each?
Solving Systems of Linear Inequalities- solve each system on a coordinate plane – clearly label boundaries as not included or included. Clearly show the OVERLAP shading. It is only the overlap that is the solution.
39) y > -2x + 2 40) 2x – 2y < 6
6x + 3y < 12 x + y 6
Applications of Systems of Inequalities
41) You need at least 6 pounds of fruit. Grapes cost $ 2 a pound and peaches cost $1.50 a pound. You cannot spend more than $18. 2 points
a) Write a system of inequalities that models this situation.
b) Graph the inequality, label the axes.
c) Find an ordered pair that is a solution to this inequality , verify that it works in each inequality and identify what it means.
UNIT 7 – EXPONENTS & EXPONENTIAL FUNCTIONS
Multiplication Properties of Exponents - simplify each completely
42) x3 • x5 43) (x3y5)(-y7x) 44) (y3)5 45) (-2k3)5
46) (3x2y4)3 47) (x3y5)(-x4y6z3) 48) (5x2y)4 49) (2a5b2)3(a2b)4
Division Properties of Exponents simplify each completely – no negative exponents in answers!
50) 51) 52)
53) 54)
Zero & Negative Exponents Properties simplify each completely – no negative exponents in answers!
55) 3y-5 56) m7 m-12 57) (3b-3)2
Simplify each completely – no negative exponents in answers!
58) 59) 60)
Mixed Properties of Exponents simplify each completely – no negative exponents in answers!
61) 62)
63) 64)
Rational Exponents
Rewrite as Rational Exponents and then evaluate. If you have a decimal result, turn it into a fraction, if that is not possible, then round to the nearest hundredth.
65) 66) 67)
Simplify Rational Exponent Expressions completely
68) x2/3 ·x4/3 69) (y1/6)3 70) 71)
72) (x1/2 · x1/3)6 73) 74)
Exponential Regression
75) A population of single-celled organisms was grown in a Petri dish over a period of 16 hours. The number of organisms at a given time is recorded in the table below. 2 points
a) Input the data into your graphing calculator and draw your graph (label axes & y-intercept)
b) Write an exponential equation for the data, rounding your “a” value to nearest whole number and your “b” value to the nearest hundredth.
c) What does the “a” value represent in the problem situation?
d) What does the “b” value represent in the problem situation?
e) Evaluate f(7) – what does this represent in the problem situation?
Exponential Growth & Decay show work on questions by showing full value from calculator, then rounding appropriately for the problem situation.
76) A business had a $5000 profit in 1990. The profit increases exponentially by 12% each year. If 1990 correspond to t = 0, then answer the following questions.
a) Write an exponential growth model for the company’s profit.
b) Use the model to find the profit in 1995.
d) Use the model to determine when the profit will double. (hint – use calc, Y1 & Y2, intersect)
77) A tire company had 14,000 employees in 1990. Each year for 10 years the number of employees decreased by 4%. Write an exponential decay model for the company’s employees.
a) Write an exponential decay model for the number of employees.
b) How many employees will the company have in 2000?
UNIT 8A – ADVANCED ALGEBRA SKILLS
Simplifying Radicals
Simplify the following expressions. Do not use decimals. Leave your answer as a radical in simplest form.
78) 79) 80) 81)
Operations with Radicals (multiply, add, subtract)
82) 83) 3 6 84) 85)
Rationalizing the Denominator
86) 87) 88) 89)
Adding & Subtracting Polynomials Add or subtract the following polynomials.
90) 91)
Multiplying Polynomials multiply the following polynomials – remember you can use BOX or the distributive property.
92) 93) (x + 9) (x – 4) 94) (x – 7)2
95) (2x + 5)2 96) (x – 3)(5x2 + 2x + 5) 97) (4X + 3)(4X – 3)
Factor the following by factoring out the GCF (upside down division)
98) 24x³ + 18x² 99) 5x3y – 15x2 100) -72w3 – 90w
Factoring Polynomials (leading coefficient of one) Factor each trinomial into two binomials, use shortcut
101) m2 + 7m +10 102) x2 + 5x – 14 103) x2 – 5x + 6
Factoring Polynomials (leading coefficient > one) Factor each trinomial into two binomials – use box method
104) 3x2 + 17x + 10 105) 18x2 + 9x – 14 106) 5x2 – 7x + 2
Factoring Special Products – difference of squares, perfect square trinomials
107) x² – 9 108) 16 – 121 x² 109) t² + 10t + 25 110) w² – 16w + 64
Factor the following by factoring out the GCF then factoring the remaining polynomial
111) 6y² - 24 112) 18 – 2x² 113) 3m3 - 27m
114) 5x³ + 30x² + 40x 115) 45x4 – 20x2 116) 3x4 + 12x3 + 9x2
Solve the following equations using the square root method
117) 4x² – 8 = 0 118) 2x² = -32 119) 2(x + 8)2 – 100 = 0
Solve the following equations using the zero product property (after factoring)
120) x2 – 8 = -7x 121) 5x² – 9x + 4 = 0 122) 9x2 – 25 = 0
Solve the Following equations using Completing the Square
123) x2 + 8x = -4 124) x2 – 2x = 24
Solve the following equations using the quadratic formula
125) x2 – 4x + 2 = 0 126) 2x2 + 2x + 2 = 0 127) 56x2 + 53x = 55
UNIT 8B – QUADRATIC FUNCTIONS
Key Features of Quadratic Functions – do these without the graphing function of your calculator
Intercept form: identify the direction of opening, the x-intercepts, the equation for the axis of symmetry, the vertex (whether it is a maximum or minimum) and the y-intercept. Also domain, range, name increasing & decreasing intervals and end behavior (both left and right)
128) y = -(x – 1)(x – 9) 129) y = x2 + 2x – 35 this one must be factored first
direction of opening______direction of opening______
x-inter______x-inter______
AOS ______AOS ______
Vertex______Vertex______
y-intercept______y-intercept______
domain______domain______
range ______range______
increasing______increasing______
decreasing______decreasing______
end behavior end behavior
left______left______
right______right______
Vertex Form: identify the direction of opening, the vertex (whether it is a maximum or minimum), the equation for the axis of symmetry, the y-intercept and the x-intercepts. Also domain, range, name increasing & decreasing intervals and end behavior (both left and right)
130) y = (x + 17)2 – 9 131) y = -1/2(x – 5)2 + 32
direction of opening______direction of opening______
Vertex______Vertex______
AOS ______AOS ______
y-intercept______y-intercept______
x-inter______x-inter______
domain______domain______
range ______range______
increasing______increasing______
decreasing______decreasing______
end behavior end behavior
left______left______
right______right______
Standard Form: identify the direction of opening, the y-intercept, the vertex (whether it is a maximum or minimum), the axis of symmetry & the x-intercepts (using the quadratic formula). Also domain, range, name increasing & decreasing intervals and end behavior (both left and right)
132) y = 2x2 – 9x + 4 133) y = -4x2 + 20x – 25
direction of opening______direction of opening______
Vertex______Vertex______
AOS ______AOS ______
y-intercept______y-intercept______
x-inter______x-inter______
domain______domain______
range ______range______
increasing______increasing______
decreasing______decreasing______
end behavior end behavior
left______left______
right______right______
134) Analyze the quadratic function equation f(x) = x2 + 3x – 24 by doing the following:
a) Draw a graph of this function in the space below
b) Label the following key features on your graph with ordered pairs:
a. Vertex
b. X-intercepts (as rounded decimals if necessary)
c. Y-intercept
d. Point symmetric to y-intercept
c) Draw the axis of symmetry as a dotted line and label with its equation
d) Is the vertex a maximum or minimum? ______
Solutions of a Quadratic Function Equation
135) The solutions of a quadratic function equation can be found with one of four possible methods. They are:
1) ______
2) ______
3) ______
4) ______
136) The solutions of the equation tell you about what specific characteristic of the graph of the quadratic?
137) There are three possible types of solutions and they tell you something specific about the graph of your quadratic. Indicate what each type tells you
a) two real solutions:
b) one real solution:
c) Two imaginary solutions
The Discriminant – the discriminant (b2 – 4ac) helps you to predict what types of solutions a quadratic function will have. In the following problems - find the discriminant, tell what it predicts, then solve the quadratic function equation with any of the three methods to verify your prediction. 2 points each