Algebra 2
Final Exam Study Guide
Name: ___________________________
Date: ______________ Block: ______
In addition to doing the problems below, also study all quizzes, tests, notes, and homework.
Final Exam Format:
45 Multiple Choice worth 1 pt each, 9 Open Ended worth 5 pts each for a Total of 90 pts.
Parent Functions
 Graph parent functions and their transformations
 Write the equations of transformed parent functions given a graph or description
 Graph and write the equation of piecewise functions.
1)
For each equation, state the name of the parent function and sketch the parent function shape.
Parent Function
Name
Sketch with important points
𝑓(𝑥) = 𝑥 3
1
𝑓(𝑥) = 𝑥 3
𝑓(𝑥) = 𝑥 2
1
𝑓(𝑥) = 𝑥 2
1
𝑓(𝑥) = |𝑥|
𝑓(𝑥) = ⟦𝑥⟧
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2)
Using the following piecewise function:
Evaluate:
𝑥 2 − 4, 𝑥 < 0
𝑓(𝑥) = { 1, 0 ≤ 𝑥 ≤ 3
|𝑥 − 4| − 2, 𝑥 > 3
𝑓(−2) =_____________________
𝑓(3) =_____________________
𝑓(6) =_____________________
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For problems 3-5, the functions do not have any vertical or horizontal stretches or compressions.
Write the equation.
3)
𝑓(𝑥) =_____________________
4)
𝑓(𝑥) =_____________________
2
5)
𝑓(𝑥) =_____________________
For problems 6-7, write the equation for each piecewise function shown.
y
y










x

6)










x


















7)
𝑓(𝑥) =
{





𝑓(𝑥) =
{
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8)
Write the equation of any transformed parent function whose range would be (−∞, 3].
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10)
Write the equation of any transformed parent function that is increasing on (−∞, ∞).
3
CHAPTER 5 Quadratic Equations and Functions
We did Chapter 5 before the midterm, but we used many of the things we learned there throughout the rest of the
course. These are the most important topics from Chapter 5, those that you’ll need to know for the Final Exam:



Standard form of a quadratic function:𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 vs. factored form
Know all types of factoring! Trinomial, GCf, Difference of 2 Squares, Housing
Solving Quadratics using the Zero Product Property: If 𝑎 ∙ 𝑏 = 0, then either 𝑎 = 0 or 𝑏 = 0
For problems 1-4, solve by factoring or taking square roots.
1)
2)
𝑥 2 − 7𝑥 = 0
3)
𝑥 2 + 2𝑥 − 8 = 0
2𝑥 2 − 6𝑥 − 8 = 0
4)
𝑥2 − 9 = 0
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For problems 5-8, factor the polynomial expression.
5)
4𝑥 2 − 25
6)
8𝑥 2 + 10𝑥 + 3
7)
𝑥 2 − 3𝑥 − 54
4
8)
−6𝑥 2 𝑦 9 + 7𝑥 3 𝑦 5
CHAPTER 6 Polynomials and Polynomial Functions




Standard form of a polynomial: 𝑃(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0 (descending order by degree)
Classify a polynomial: By degree and number of terms
o Names of degrees 0 through 5: Constant, linear, quadratic, cubic, quartic, quintic
o Names for 1 through 4 terms: Monomial, binomial, trinomial, polynomial of 4 terms
Factor Theorem: The expression (𝑥 − 𝑎) is a factor of a polynomial if and only if a is a zero of the related
polynomial function.
Polynomial long division and synthetic division
𝑝
Possible rational roots of a polynomial in standard form:𝑞 , where p is a factor of 𝑎0 and q is a factor of 𝑎𝑛 .
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Conjugates / Complex Conjugates: 𝑎 + √𝑏and 𝑎 − √𝑏; 𝑎 + 𝑏𝑖 and 𝑎 − 𝑏𝑖
Irrational Root Theorem: If 𝑎 + √𝑏 is a root, so is 𝑎 − √𝑏
Imaginary Root Theorem: If 𝑎 + 𝑏𝑖 is a root, so is 𝑎 − 𝑏𝑖
What this means: A polynomial must have an even number of irrational / imaginary roots.
An nth degree polynomial has this many roots: n (number of roots = biggest exponent)
Know how to determine the end behavior, find the zeros, and graph.
Know how to write the equation given the zeros.

For problems 1-2, write each polynomial in standard form. Then classify it by degree and by number of terms.
1)
2)
𝑥 − 𝑥3 − 𝑥5
3𝑥 + 2𝑥 2 − 𝑥 + 4𝑥 3
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For problems 3-4, write the polynomial in factored form. List the zeros of the function and their multiplicity. Also
state the end behavior.
3)
𝑓(𝑥) = 𝑥 3 − 𝑥 2 − 12𝑥
4)
𝑓(𝑥) = 𝑥 3 (𝑥 + 2)4
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5)
Write a polynomial function in standard form with the given zeros. 𝑥 = 1, 3 with multiplicity of 2.
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6)
State the degree, leading coefficient, and the type of end behavior (informal up – up etc.).
Polynomial
𝑓(𝑥) = −3𝑥 6 − 2𝑥 3 + 𝑥 − 8
Degree
Leading Coefficient
5
Type of end behavior
7)
Divide. Use both long division and synthetic division. (𝑥 3 + 3𝑥 2 − 2𝑥 − 4) ÷ (𝑥 − 2)
____________________________________________________________________________________________________________________________________
8)
Use synthetic division and the given factor to completely factor the following.
𝑥 3 − 3𝑥 2 − 𝑥 + 3
Known factor: 𝑥 + 1
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9)
Use the Remainder Theorem to find if x-3 is a factor of P (x )  x 4  x 3  x 2  2x . Explain why or why not.
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10)
Solve for 𝑥.
𝑥 3 − 5𝑥 2 + 4𝑥 = 0
6
11)
State the number of complex roots, the possible number of real roots, and the possible rational roots. Then
find all the roots.
𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 𝑥 + 5
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12)
Simplify:
16𝑥 8 𝑦 −2 𝑧 12
(2𝑥𝑦 3 𝑧)3
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13)
Find the perimeter of the following figure. Simplify your answer!
5𝑥 2 + 2𝑥 − 3
2𝑥 + 3
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14)
Find the area of the following figure. Simplify your answer!
5𝑥 2 + 2𝑥 + 1
2𝑥 − 4
7
15)
Name the degree, number of solutions, number of real solutions, and number of imaginary solutions.
y





Degree
_______
# of Solutions
________
# of Real Solutions
_______
# of Imaginary Solutions
________

x
















4
𝑓(𝑥) = 𝑥 − 4𝑥 2 − 9𝑥 − 18


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26)
Divide using synthetic division:
(4𝑥 5 + 6𝑥 4 − 6𝑥 3 + 𝑥 2 − 3𝑥 + 1) ÷ (𝑥 + 2)
8
CHAPTER 7 Radical Functions and Rational Exponents
𝑛



Definition of nth root: If 𝑎𝑛 = 𝑏, a is the nth root of b, or √𝑏 = 𝑎.
Know how to find real roots (using calculator) and simplify radical expressions.
𝑛
𝑛
𝑛
Multiply radicals: √𝑎 ∙ √𝑏 = √𝑎𝑏

Divide radicals: 𝑛 = √𝑏


To add/subtract radicals: They must be like radicals (same index and same radicand).
To multiply binomial radical expressions: Use the FOIL method.


Rational exponents: 𝑎 𝑛 = √𝑎𝑚 = ( √𝑎)
Properties of rational exponents:
(𝑎𝑚 )𝑛 =𝑚𝑛
o 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛
𝑛
√𝑎
√𝑏
𝑛
𝑚
o
𝑎𝑚
𝑎𝑛
= 𝑎𝑚−𝑛
𝑎
𝑛
𝑛
𝑚
(𝑎𝑏)𝑚 = 𝑎𝑚 𝑏 𝑚
𝑎 𝑚
1
𝑎−𝑚 = 𝑎𝑚
𝑎𝑚
(𝑏 ) = 𝑏𝑚
For problems 1-2, find the indicated root if possible.
1)
√−144
4
2)
√𝑥 8
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For problems 3-8, simplify each expression. Assume all variables are positive.
3)
√24𝑥 14 𝑦 3 𝑧 7
3
4)
2√536𝑧 80
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5)
√5𝑥 4 𝑦 3 ∙ √45𝑥 3 𝑦
3
√56𝑦 5
6)
3
√7𝑦
9
7)
3
8)
√27 + √75 − √12
3
3
5√24 + 18√81 − 10√192
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9)
2
Write the expression in radical form. 𝑥 3
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For problems 10-11, simplify the radical expression.
10)
4
2435
11)
10
1
2
𝑥 6𝑥 3
CHAPTER 9 Rational Functions
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To simplify a rational expression:
o Factor the numerator and denominator.
o State restrictions based on the denominator. (denominator can not equal zero)
o Cancel out any common factors.
When multiplying rational expressions, combine together the numerators and denominators, then
simplify as described above.
When dividing rational expressions, multiply by the reciprocal of the second fraction. The restrictions are
based on both denominators and the numerator of the second expression
To add or subtract rational expressions:
Factor the denominators, then find a common denominator.
Multiply the fractions by any missing factors to create the common denominator.
Add or subtract the fractions.
For problems 1-14, simplify each rational expression. State any restrictions on the variable.
1)
𝑥 2 −2𝑥−24 𝑥 2 −1
∙
𝑥 2 +7𝑥+12 𝑥−6
2)
4𝑥 2 −2𝑥
2𝑥
÷ 𝑥 2 +2𝑥+1
𝑥 2 +5𝑥+4
3)
3𝑥
6
+ 𝑥+2
𝑥 2 −4
4)
1
2
− 𝑥 2 +3𝑥
𝑥 2 −2
5)
𝑥 2 +6𝑥+5
𝑥 2 −25
6)
6𝑥−18 𝑥 2 +5𝑥+6
∙ 𝑥 2 −9
𝑥 2 −4
11
7.)
4𝑥 3 +2𝑥 2
𝑥 2 +15𝑥+14
2𝑥
÷ 𝑥 2 −7𝑥−8
9)
4𝑥+3
5
− 2𝑥−1
8𝑥+1
11)
2
𝑥
3
3+
𝑦
2+
8)
𝑥
𝑥+5
10)
7
2𝑥−5
− 𝑥 2 −13𝑥+36
𝑥−9
12)
+
𝑥+10
𝑥+5
2
𝑥
6
4−
𝑥
1+
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For problems 13-14, solve the rational equation.
1
𝑥
18
13)
+6= 𝑥
2
14)
12
1
2
𝑥
3
+6=𝑥
CHAPTER 8 Exponential and Logarithmic Functions
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
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1)
Form of an exponential function:𝑦 = 𝑎 ∙ 𝑏 𝑥 . Growth factor: 𝑏 > 1; Decay factor: 𝑏 < 1
To find the inverse of a function:
o Switch x and y.
o Solve for 𝑦
Solving a logarithmic and exponential equation.
Write in logarithmic form.
2−3 = 0.125
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2)
Evaluate.
log 2 64
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3)
Find the inverse of the function.
𝑓(𝑥) = √𝑥 + 2
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4)
Find the inverse of the function.
𝑓(𝑥) = 2(𝑥 + 1)3 + 5
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For problems 5-8, solve each equation. Round to the nearest hundredth.
5)
5𝑥 = 9
6)
7𝑥−3 = 25
7)
log 3𝑥 = 1
13
8)
2 log 3 𝑥 = 54