Algebra 2 Final Review
Study Outline
• Chapter 5:
• Chapter 6:
• Chapter 7&9:
• Chapter 11:
• Chapter 13:
• Chapter 8:
Quadratic Functions
Polynomial Functions
Functions and Inverses
Exponential and
Logarithmic Functions
Statistics
Trigonometry
Rational Functions
Chapter 5: Quadratic Functions
Standard Form: 𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Vertex Form: 𝑓 𝑥 = 𝑎(𝑥 − ℎ)2 +𝑘 with Vertex (h,k)
Axis of Symmetry: 𝑥 =
𝑏
−
2𝑎
Ex. Find the axis of symmetry for the parabola given
by 𝑓 𝑥 = −𝑥 2 + 8𝑥 − 9.
Vertex
The maximum or minimum point of a parabola.
𝑏
−
2𝑎
From standard form, use 𝑥 =
to find the 𝑥coordinate and then plug that into the function to
find the 𝑦 −coordinate.
You can also find the vertex by graphing the
function and using the minimum/maximum feature
on your calculator.
Ex. Find the vertex of 𝑓 𝑥 = 𝑥 2 − 4𝑥 + 7.
Transformations
𝑓 𝑥 =𝑎 𝑥−ℎ
2
+𝑘
Ex. Write a function for a parabola that is
reflected across the 𝑥 axis, vertically stretched
by a factor of 5, and translated 3 units to the
right and 9 units down.
Ex. Describe the transformations given by the
1
function 𝑓 𝑥 = 𝑥 + 3 2 + 9 .
4
Solving Quadratic Equations
•
•
•
•
•
Graphing
Factoring
Square-Rooting
Completing the Square
Quadratic Formula
Quadratic Formula
Used to find roots of quadratic equations.
𝑥=
−𝑏± 𝑏2 −4𝑎𝑐
2𝑎
𝐷𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 = 𝑏 2 − 4𝑎𝑐
Used to determine the number and types of
roots of a given quadratic equation.
Note: roots = zeros = solutions = x-intercepts
Quadratic Formula
Find the roots of 𝑥 2 − 2𝑥 + 5 = 0
Discriminant
𝐷𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 = 𝑏 2 − 4𝑎𝑐
Ex. Find the discriminant of the function 𝑓 𝑥 =
− 2𝑥 2 + 5𝑥 − 9.
Based on the discriminant, determine the
number and types of roots of the function.
Completing the Square
Find the value of 𝑐 to complete the square,
creating a perfect square trinomial.
Ex. 𝑥 2 − 8𝑥 + 𝑐
Irrational Numbers
𝑖 = −1
𝑖 2 = −1
Ex. Simplify (3 + 2𝑖)(2 − 4𝑖)
Chapter 6: Polynomial Functions
Characteristics of a polynomial:
 Degree: Highest power
 Leading Coefficient: Coefficient of term with
highest power
 Number of Terms
Ex. 𝑓 𝑥 = −8𝑥 5 + 2𝑥 2 − 7𝑥 9
End Behavior
Step 1: Leading coefficient (right side)
 Positive: Right side rises
 Negative: Right side falls
Step 2: Degree (both ends)
 Even: both ends go in the SAME direction
 Odd: the ends go in OPPOSITE directions
End Behavior
Ex. Describe the end behavior of the function:
𝑓 𝑥 = −3𝑥 7 + 5𝑥 3 − 2𝑥 = 10
Add/Subtract/Multiply
Adding  combine LIKE terms
Subtracting  remember to distribute the –
Ex. 3𝑥 5 − 9𝑥 2 − 10 − (2𝑥 5 − 3𝑥 4 + 5𝑥 2 )
Ex. (𝑥 − 3)(𝑥 2 + 2𝑥 − 1)
Solving Polynomial Equations
To solve ALGEBRAICALLY:
• Factor completely
• Set each factor = 0 and solve for x
Ex. 5𝑥 4 − 10𝑥 2 = 0
Note: Remember that the degree tells you how
many total solutions there are.
Solving Polynomial Equations
To solve GRAPHICALLY, find the x-intercepts:
Ex. 3𝑥 4 + 24𝑥 3 + 48𝑥 2 = 0
Multiplicity of Roots
• On a graph, we can tell the multiplicity of a
root because an even multiplicity will
“bounce” off the 𝑥-axis while an odd
multiplicity will “bend” through the 𝑥-axis.
• Find the roots and state the multiplicity of
each for 𝑓 𝑥 = 𝑥 4 𝑥 − 1 3
Chapters 7 & 9:
Functions & Inverses
Exponential Functions
Find the inverse:
Ex. 𝑓 𝑥 =
𝑥+6
2
Composition of Functions
Use the answer from one function as the input
in the other function.
Ex. Given 𝑓 𝑥 = −3𝑥 − 5 and 𝑔 𝑥 = 𝑥 2 , find:
𝑓 𝑔 2
𝑔(𝑓 2 )
Exponential Functions:
Growth and Decay
𝑓 𝑥 = 𝑎 ∙ 𝑏𝑥
 Growth when 𝑏 > 1
 Decay when 0 < 𝑏 < 1
 𝑎 is 𝑦-intercept
Ex. Determine the y-int, base, growth or decay
𝑓 𝑥 = 3 1.03 𝑥
𝑔 𝑥 = 0.95𝑥
Exponential Growth and Decay
𝐴=𝑃 1±𝑟
𝑡
Ex. An investment of $4,250 is said to gain value
at 4% annually. How long will it take the
investment to be worth $6000?
Solving Exponential Equations
Solve:
Ex. 165𝑥 = 64𝑥+7
Ex. 25𝑥 =
1 2−𝑥
125
Logarithmic Expressions
Definition of log: log 𝑏 𝑎 = 𝑝 ↔ 𝑏 𝑝 = 𝑎
Rewrite each expression in the “other” form:
3
−2
=
1
9
log 4 64 = 3
Logarithmic Properties
• Product Rule: log 𝑏 𝑥 + log 𝑏 𝑦 = log 𝑏 𝑥 ∙ 𝑦
• Quotient Rule: log 𝑏 𝑥 − log 𝑏 𝑦 =
• Power Rule: log 𝑏 𝑥 𝑝 = 𝑝 ∙ log 𝑏 𝑥
𝑥
log 𝑏
𝑦
Logarithmic Properties
• Express the following as a single logarithm.
• log 4 32 + log 4 2
Solving Exponential or Logarithmic
Equations
• When it is not possible to get “like bases” take
the log of both sides, rewrite the equation and
solve.
• Ex. 4𝑥 = 18
Solving Exponential or Logarithmic
Equations
• A logarithmic equation:
– Condense
– Rewrite as an exponential equation
– Solve
• Ex. log 2 4 + log 2 𝑥 = 6
Logarithmic Functions
• Logarithmic functions: 𝑓 𝑥 = log 𝑏 𝑥
– Domain: 𝑥 > 0
– Range: All Real Numbers
– 𝑥 −intercept: 𝑥 = 1
– 𝑦 −intercept: None
Chapter 11: Statistics
• Linear Regression and Exponential Regression
– Enter the data into L1 and L2
– Stat -> Calc -> 4: LinReg(ax+b)
• 𝑦 = 𝑎𝑥 + 𝑏
– Stat -> Calc -> 0: ExpReg
• 𝑦 = 𝑎 ∙ 𝑏𝑥
Exponential Model
• Given the following data, find the equation of
the exponential model.
Years after 1970
Population (in millions)
0
203.3
10
226.5
20
248.7
30
281.4
Correlation
• The correlation, 𝑟, measures the strength and
direction of a linear relationship.
• 𝑟 close to 1 OR -1 is strong
• 𝑟 close to 0 is weak.
Mean
• Find the mean from the frequency table:
Score
Frequency
90
3
92
8
95
11
99
2
Standard Deviation
• Standard deviation measures the “spread” or
“variability” of the data.
• A small standard deviation indicates data that
are all very similar.
• A large standard deviation indicates data that
are very different.
Normal Distribution
• Remember the 68 – 95 – 99.7% rule
Normal Distribution
• Test scores were normally distributed with a
mean of 100 and standard deviation of 10.
What percent of students scored between an
80 and 120?
Types of Studies
• Survey: a questionnaire given to a sample of
individuals.
• Observational Study: gather data on a topic
without manipulating any variable
• Experiment: purposefully manipulate one
variable to examine its effect on another
variable
Types of Studies
• Categorize each type of study:
• A college sends a feedback postcard to students
who recently attended an open house.
• A researcher compares the SAT scores of students
taking Latin with students not taking Latin.
• The number of heart attacks is compared when
one group of individuals are assigned to take an
aspirin a day while another group does not not.
Chapter 13: Trigonometry
• SOH-CAH-TOA
• Given the triangle, find the value of the three
trig functions.
Angle of Elevation or Depression
• A road rises 10 feet over a horizontal distance
of 80 ft. Find the angle of elevation of the
road.
Angles of Rotation
•
•
•
•
Positive = Counterclockwise
Negative = Clockwise
Draw each angle in standard position:
𝜃 = 210°
𝜃 = −100°
Coterminal Angles
• Coterminal angles are in the same position.
• Add or subtract multiples of 360°.
• Find two angles coterminal with 𝜃 = 40°.
Reference Angles
• Think reference to the 𝑥-axis.
• Find the reference angle for each angle.
• 𝜃 = 200°
𝜃 = 300°
Points on the Terminal Side of 𝜃
• Point P(5, -12) is on the terminal side of 𝜃
when drawn in standard position. Find the
values of sin 𝜃, cos 𝜃, and tan 𝜃.
Radians and Degrees
• 180° = 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
• Convert to the “other” measurement.
• 210°
5𝜋
6
Exact Value or Unit Circle
sin 𝜃
cos 𝜃
tan 𝜃
𝟑𝟎°
𝟒𝟓°
𝟔𝟎°
1
2
3
2
3
3
2
2
2
2
3
2
1
2
1
3
Exact Value of Sin, Cos, Tan
• Find the reference angle. (Convert to degrees
if necessary.)
• Use the table of exact values.
• Decide if the function should be positive or
negative by using A-S-T-C.
Exact Value of Sin, Cos, Tan
• Find the exact value of:
sin 210°
7𝜋
cos
4
Exact Value of Sin, Cos, Tan
• Find the value of the angle that is the solution
to: tan
−1
3
3
Pythagorean Identity
• Remember that sin2 𝜃 + cos2 𝜃 = 1
Graphs of Sine/Cosine
• Amplitude = |a|
• Graph 𝑦 = 2 sin 𝑥
Period =
2𝜋
𝑏
Graphs of Sine/Cosine
• Amplitude = |a|
• Graph 𝑦 = cos 𝜋𝑥
Period =
2𝜋
𝑏
Maximum and Minimum Value for
Sine and Cosine
• Given 𝑦 = 𝑎 sin 𝑏𝑥 + 𝑘, the max or min values
can be found by: 𝑘 ± 𝑎.
• Find the max/min values of each function:
𝑦 = 2 sin 3𝑥
𝑦 = 3 cos 𝜋𝑥 + 5
Chapter 8: Rational Expressions and
Equations
• Simplify by factoring, then canceling factors in
common.
9𝑥 2 𝑦
10𝑥𝑦 3
∙
5𝑥
3𝑦
𝑥 2 −4
3𝑥−9
𝑥+2
÷
𝑥−3
Restricted Values of 𝑥
• Remember that 0 cannot be in the
denominator!
• Simplify and state the restricted values of 𝑥
4𝑥 + 4
𝑥 2 − 8𝑥 − 9
Solving Rational Equations
• Determine the LCD. Multiply EACH term by
the LCD. Simplify and solve. Check for
extraneous solutions.
• 𝑥
12
+
𝑥
=7
Solving Rational Equations
•
𝑥+4
𝑥
+
𝑥−6
2
=
10
𝑥−6