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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.