Algebra 2 Final Review

Study Outline

• Chapter 5: Quadratic Functions• Chapter 6: Polynomial Functions• Chapter 7&9: Functions and Inverses

Exponential and Logarithmic Functions

• Chapter 11: Statistics• Chapter 13: Trigonometry• Chapter 8: Rational Functions

Chapter 5: Quadratic Functions

Standard Form:

Vertex Form: with Vertex (h,k)

Axis of Symmetry:

Ex. Find the axis of symmetry for the parabola given by .

Vertex

The maximum or minimum point of a parabola. From standard form, use to find the - coordinate and then plug that into the function to find the 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 .

Transformations

Ex. Write a function for a parabola that is reflected across the 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 function + 9 .

Solving Quadratic Equations• Graphing• Factoring• Square-Rooting• Completing the Square• Quadratic Formula

Quadratic FormulaUsed to find roots of quadratic equations.

Used to determine the number and types of roots of a given quadratic equation.

Note: roots = zeros = solutions = x-intercepts

Quadratic FormulaFind the roots of

Discriminant

Ex. Find the discriminant of the function .

Based on the discriminant, determine the number and types of roots of the function.

Completing the Square

Find the value of to complete the square, creating a perfect square trinomial.

Ex.

Irrational Numbers

Ex. Simplify

Chapter 6: Polynomial Functions

Characteristics of a polynomial: Degree: Highest power Leading Coefficient: Coefficient of term with

highest power Number of Terms

Ex.

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:

Add/Subtract/Multiply

Adding combine LIKE termsSubtracting remember to distribute the –

Ex.

Ex.

Solving Polynomial Equations

To solve ALGEBRAICALLY:• Factor completely• Set each factor = 0 and solve for x

Ex.

Note: Remember that the degree tells you how many total solutions there are.

Solving Polynomial EquationsTo solve GRAPHICALLY, find the x-intercepts:Ex.

Multiplicity of Roots

• On a graph, we can tell the multiplicity of a root because an even multiplicity will “bounce” off the -axis while an odd multiplicity will “bend” through the -axis.

• Find the roots and state the multiplicity of each for

Chapters 7 & 9: Functions & Inverses

Exponential FunctionsFind the inverse:Ex.

Composition of Functions

Use the answer from one function as the input in the other function.

Ex. Given and , find:

Exponential Functions:Growth and Decay

Growth when Decay when is -intercept

Ex. Determine the y-int, base, growth or decay

Exponential Growth and Decay

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. Ex.

Logarithmic Expressions

Definition of log: Rewrite each expression in the “other” form:

Logarithmic Properties

• Product Rule:

• Quotient Rule:

• Power Rule:

Logarithmic Properties

• Express the following as a single logarithm.

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.

Solving Exponential or Logarithmic Equations

• A logarithmic equation:– Condense– Rewrite as an exponential equation– Solve

• Ex.

Logarithmic Functions

• Logarithmic functions: – Domain:

– Range: All Real Numbers

– intercept:

– intercept: None

Chapter 11: Statistics

• Linear Regression and Exponential Regression– Enter the data into L1 and L2

– Stat -> Calc -> 4: LinReg(ax+b)

– Stat -> Calc -> 0: ExpReg

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, , measures the strength and direction of a linear relationship.

• close to 1 OR -1 is strong

• 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:•

Coterminal Angles

• Coterminal angles are in the same position.

• Add or subtract multiples of .

• Find two angles coterminal with .

Reference Angles

• Think reference to the -axis. • Find the reference angle for each angle.•

Points on the Terminal Side of

• Point P(5, -12) is on the terminal side of when drawn in standard position. Find the values of , , and .

Radians and Degrees

• Convert to the “other” measurement.•

Exact Value or Unit Circle

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:

Exact Value of Sin, Cos, Tan

• Find the value of the angle that is the solution to:

Pythagorean Identity• Remember that

Graphs of Sine/Cosine

• Amplitude = |a| Period = • Graph

Graphs of Sine/Cosine

• Amplitude = |a| Period = • Graph

Maximum and Minimum Value for Sine and Cosine

• Given , the max or min values can be found by: .

• Find the max/min values of each function:

Chapter 8: Rational Expressions and Equations

• Simplify by factoring, then canceling factors in common.

Restricted Values of

• Remember that 0 cannot be in the denominator!

• Simplify and state the restricted values of

Solving Rational Equations

• Determine the LCD. Multiply EACH term by the LCD. Simplify and solve. Check for extraneous solutions.

•

Solving Rational Equations

𝑥+4𝑥−6

+𝑥2=10𝑥−6