Unit 1 Review

Parallelogram Properties

Remember, you have to be able to prove these properties using coordinates, congruent triangles, or parallel lines and apply them to solve problems!

All parallelograms have:

Opposite sides _______________ and _________________

Opposite angles __________________

Diagonals that _______________ each other

Diagonals that divide the parallelogram into ________________________

Formulas to help: Slope = Distance = Midpoint =

Volume Formulas

Volume of a right prism (including a rectangular prism) or cylinder = Area of __________ • ______________

Volume of a cone or pyramid = Areaof ¿3

¿

Volume of a sphere = 4Π r3

3

When using area or volume to calculate density, _____________ what you are measuring by the area or volume to determine the density per square (area) or cubic (volume) unit.

When using formulas to maximize volume, you can set up the volume formulas using the given parameters and use technology to find the maximum value. The steps in a TI-84 are:

1. Graph the formula in __________

2. Adjust the _____________ to see the minimum or maximum value for the appropriate domain

3. Push 2nd - Trace (Calc) - #3 minimum or #4 maximum, depending on the question asked

4. Set your ________________ to the right and left of the minimum or maximum point, then get your value

Concept Questions:

1. What could the slope, midpoint, or distance formulas tell us about triangles or parallelograms?

2. What are some real-world applications of density? Why are they important?

Unit 1 Sample Problems1. 2.

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5. 6.

Unit 2 ReviewEquation of a Circle

(x – h)2 + (y – k)2 = r2

(h, k) = ____________________ r = _______________________Example: Find the equation, center, and radius of the circle:

x2 + y2 + 16x – 6y = 8

CirclesArea of a Circle = _________ Circumference of a Circle = ____________

Area of a SECTOR of a circle = ___________________

Arc Length of a Sector of a circle = ______________________

Tangent lines, lines ___________ the circle that touch it at ___________________, form

a right angle with the _____________ it intersects.

A radius that _________________ a chord forms a right angle with the chord.Radian Angle Measure

Radians = ______________ Degrees = _________________

Examples: Convert 5π4 radians to degrees. Convert 3000 to radians.

Radian Measure in Circles: Arc Length = __________________ • _______________

Angle-Arc Relationships

Central Angle - ______________ measure of intercepted arc, vertex at the ____________

Inscribed Angle - ______________ measure of intercepted arc, vertex on the _____________

Circumscribed Angle - Angle formed by _____________ lines to circle

Circumscribed Angle

Circle Segments

Chord - Segment connecting two points _________________

Tangent - Segment that touches a circle in ________________

Secant - Line/segment that intersects a circle in __________________

Secant-Tangent Intersect Two Chords Intersect Two Secants Intersect

Conceptual Questions:

1. How does the Pythagorean Theorem relate to the equation of a circle?

2. Draw a triangle inside the circle below with all its vertices on the circle. What kind of angles are the vertices? Why is an inscribed angle half the measure of the intercepted arc?

3. Why do we divide the degrees by 360 to compute the arc length or area of a sector of a circle?

Unit 2 Sample Problems1. 2.

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Unit 3 ReviewStatisticsExperiment – Study comparing a _____________ group and an _______________ group

Observation – Study observing the characteristics of __________ group

Simple Random Sample – Subjects are chosen from a population _________________

Systematic Random Sample - Subjects are chosen ______________________________

Convenience Sample - Subjects are chosen __________________________________

Stratified Random Sample - Subjects are chosen ________________________________

Mean (x) – Statistical ______________

Standard Deviation (σ) – Amount which data _____________ from the mean

Margin of Error - Expected value that a sample mean could deviate from the actual ____________________

Margin of Error Formula: Standard Deviation

√n (where n is the sample size)

Bias - Unintended feelings or actions that skew data

Concept Questions:

1. Why is it important that bias is limited and samples are random in a statistical study?

2. What is more important in evaluating data, the mean or standard deviation? Why?

Unit 3 Sample Problems1.

2.

3. A reporter for the school newspaper asked 75 randomly selected students if they would be traveling over spring break. Students were asked what form of transportation they would be using to travel. The school has 480 total students. Based on the results shown in the table, how many of the school’s students should be expected to travel by plane over spring break?

A) 10 B) 13 C) 64 D) 210

4. Tanner and Robbie discovered that the means of their grades for the first semester in Mrs. Merrell’s mathematics class are identical. They also noticed that the standard deviation of Tanner’s scores is 20.7, while the standard deviation of Robbie’s scores is 2.7. Which statement must be true?

A) In general, Robbie’s grades are lower than Tanner’s grades.B) Robbie’s grades are more consistent that Tanner’s grades.C) Robbie had more failing grades during the semester than Tanner had.D) The median for Robbie’s grades is lower than the median for Tanner’s grades.

5. A school newspaper will survey students about the quality of the school’s lunch program. Which method will create the least biased results?

A) Twenty-five vegetarians are surveyed.B) Twenty-five students are randomly chosen from each grade level.C) Students who dislike the school’s lunch program are chosen to complete the survey.D) A booth is set up in the cafeteria for students to voluntarily complete the survey.

6. A survey of a random sample of voters in a North Carolina Senate race predicts that candidate A will receive 52% of the votes and that candidate B will receive 48% of the votes. The margin of error is ± 3%. Based on the polling results, who will win the election? A) Candidate A wins the election.B) Candidate B wins the election.C) The candidates tie.D) All of the above are possible.

Unit 4 ReviewDefinition of Inverse Functions

Inverse functions have the _________________ and ____________________ values switched. Their graphs are

a reflection over the ___________ line.

To find an inverse function, _______________________ and switch the _________________.

Example 1: Find the inverse of f(x) = (x + 4)3 – 6.

Example 2: Find the inverse of f(x) = 4x – 2 – 1.

Restricting the Domains to Create Inverse Functions

For a relation to be a function, every _____________ produces one _________________.

Sometimes, a function will produce an inverse that is not a function. In that case, the domain must be restricted

so the inverse is also a function. One example of this is with quadratic functions, whose inverse is a

________________________ function.

Example 3: Find the inverse of f(x) = x2 - 1, and determine the domain on which the inverse exists.

Inverses of Exponential Functions

Exponential functions, with a _________________ as the exponent, use an operation called ______________

to determine the inverse.

To convert exponents to logarithms:

bx = a → logb a = x 10x = a → log a = x ex = a → ln a = xConcept Questions:

1. How are inverse functions and inverse operations related?

2. Why can there never be a logarithm of a negative number? (Why, in the above examples, will a never = 0?)

Unit 4 Practice Problems1. 2.

3.

Assuming f(x) represents the number of families as a function of the year, what is f-1 (30,100)?

A) 65 B) 70 C) 75 D) Not enough information

Use the following graph of f(x) for questions #4 and 5.

4) If the minimum of the function is (-0.5, -6.5),

what domain would produce an inverse function?

A) x ≤ -0.5 B) x ≥ -0.5 C) x ≤ -6.5 D) x ≥ -6.5

5) What is f-1 (-3)?

A) -6.5 B) -3 C) -0.5 D) 1

Unit 5 ReviewExponential Functions

y = abx (Growth or Decay) A = Pert (Growth compounding CONTINUOUSLY)

y = ________________ a = _______________ A = ______________ P = ________________

b = ________________ x = ______________ e = _____ r = ____________________ t = ___________

If a value increases or decreased by a percentage, the growth or decay factor is represented by ______ or _____,

where r is the percent increasing or decreasing as a decimal.

Solving Exponential Equations for the Exponent

To solve exponential equations to determine the exponent:

1. If the bases are equal, set the _______________ equal and solve.

2. If only one side has an exponent, isolate the __________ and _______________.

3. Convert the exponent to a _________________ (the inverse of an exponent).

4. Use the ______________________ formula if necessary to evaluate the logarithm

5. If the variable is not isolated, finish solving the equation.

NOTE: Exponential equations, like all other equations, can be solved by graphing the expressions on both

sides of the equation and finding the ______________________________ using technology.

Change of Base Formula

logb x = log xlog b = ln x

ln b

We can evaluate log and ln, the common logarithms, using the calculator to get a decimal approximation.

Graphs of Exponential Functions

y = abx a = _________________ b = __________________

For b > 1, graph ____________________. For 0 < b < 1, graph ____________________.

For increasing exponential functions, they will ultimately increase ________________ than other functions as x increases.

Concept Questions:

1. Why do we add or subtract 1 when determining the growth factor from a percent increase or decrease?

2. Will an exponential decay function ever equal 0? Why or why not?

Unit 5 Practice Problems1.

2.

3. 4.

5. 6. Two population functions are graphed on the same plane to

compare their growth. The first, f(x), represents one town’s

population of 20,000 growing at a 5% annual rate. The second,

g(x), represents another town’s population of 24,000 growing at a

4% annual rate. What statement applies to the y-intercepts of the

functions?

A) The y-intercept of f(x) is 4000 lower than g(x).

B) The y-intercept of f(x) is 4000 higher than g(x).

C) The y-intercept of f(x) is 1% lower than g(x).

D) The y-intercept of f(x) is 1% higher than g(x).

Unit 6 ReviewAbsolute Value Functions

The absolute value of a number, represented ______, represents its distance from zero on a number line.

It is always ______________.

The graph of an absolute value function is in the shape of a _______, because negative inputs in the domain have positive outputs. They follow the same transformation rules as other functions.

Translations:

Reflections (Flips):__________________________________________________________________________

Stretches/Shrinks: If a > 1, _______________________. If a < 1, ________________________________.

Absolute value functions can be graphed in the calculator using _________________. To solve systems

involving these equations, find the ______________________ with the other equation.

Step Functions

Greatest Integer Function - For any x-value, the y-value is the ___________ integer less than or equal to x.

It can be graphed using the following steps:

Least Integer Function - For any x-value, the y-value is the ___________ integer less than or equal to x.

It can be graphed using the following steps:

Piecewise Functions

Inside Function Outside Function

Positive (+)

Negative (–)

Piecewise functions are functions with different function rules for different ________________.

For example, they can be represented: f(x) = {−2 x , for x≤02x , for 0<x<5x2 , for x ≥5

The domain is usually continuous, but the range is not necessarily continuous depending on the values.

When evaluating piecewise functions, ________________ the input into the appropriate rule for its domain.

When graphing piecewise functions, graph each function rule for its appropriate domain.

Use _________ circles for < and >, and use ____________ circles for ≤ and ≥.

Building New Functions (Operations With Functions)

Functions can be added, subtracted, multiplied, and divided to create new functions following the same rules that apply to other expressions.

The domain of the new functions can change, however, if the operation creates ________________________ (such as 0 under a fraction bar or negatives under a radical) in the new function.

Examples:

For f(x) = 3x + 6 and g(x) = x + 2,

a) What is f(x) + g(x)? What is its domain?

b) What is f(x)/g(x)? What is its domain?

Concept Questions:

1. How is an absolute value function the same as a piecewise function?

2. Consider the functions: f(x)= 4x+9 and g(x)= -2x - 4

Evaluate f(-3). Evaluate g(-3).

Add f(x) + g(x). Evaluate (f + g)(-3).

What do you notice? What properties have you learned that explain your answer?

Unit 6 Practice Problems1. 2.

3.

4. 5. For f(x) = 2x2 + 8x, g(x) = 2x, and h(x) = x + 4, what is

f(x) - g(x) • h(x)?

A) 0 B) 1 C) 2x3 + 14x2 + 24x D) 9x2 + 36x

Unit 7 ReviewKey Terms

Degree - ________________________________________________________________________________

Leading Coefficient - ______________________________________________________________________

Solution - _______________________________________________________________________________

Graphs of Polynomials

Graphs of polynomials follow many of the same patterns as other graphs.

x-intercepts (also ______________, ________________, ______________):

y-intercepts: ___________________________________________________

Relative Minimum/Maximum Values: where graph changes direction, can be found using technology -

______________________, _____________________, _____________

End Behavior as x approaches ∞ and -∞:

Dividing Polynomials

Long Division - To divide any polynomial by another polynomial.

Synthetic Division - Shortcut to divide a polynomial by a binomial (with leading coefficient of 1)

Odd Degree Even Degree

Positive Leading Coefficient As x → - ∞, y __________As x → ∞, y ____________

As x → - ∞, y __________As x → ∞, y ____________

Negative Leading Coefficient As x → - ∞, y __________As x → ∞, y ___________

As x → - ∞, y __________As x → ∞, y ___________

Remainder Theorem - When a polynomial is divided by a binomial (x - k), the remainder is equal to the

__________________ at f(k).

Factor Theorem - When a polynomial is divided by a binomial (x - k) and the remainder is 0, (x - k) is a

__________________ of the polynomial. Therefore, the solution for x to the equation x - k = 0 is a

__________________ of the polynomial.

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra - The number of solutions, real or complex, to any function is equal to its

_________________.

Building Polynomials from Roots

If we know the solutions to polynomial functions, we can write _______________ equal to 0 that associate with the roots. Then, using these points and one more point, we can write the function.

Example: What function has roots -2, 4, and 5 and passes through point (1, 9).

We can also use the calculator’s regression feature to build these functions. Using the example above:

1. STAT-EDIT, use points ________, ___________, ___________, __________

2. STAT-CALC-CubicReg (because the function has _____ solutions)

3. Substitute the coefficients to get the function: __________________________________________

Solving Systems With Polynomials

To solve systems of equations with polynomials and other functions, using technology to find the

___________________________ is usually the most efficient way. Graph both equations on the same

coordinate plan, and determine what points satisfy both equations.

Concept Questions:

1. How do lines and parabolas relate to the end behavior of all polynomial functions?

2. Why does the Remainder Theorem guarantee that dividing a polynomial by a binomial to produce a remainder of zero proves that the solution to the binomial is a solution to the polynomial?

Unit 7 Practice Problems1. 5.

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3. 6.

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Unit 8 ReviewOperations With Rational Expressions

Rational expressions follow the same arithmetic rules as _____________________.

Multiply rationals - Factor first, then ___________________________, then divide out common factors

Dividing rationals - Factor first, then ______________________________, then divide out common factors

Adding or subtracting rationals - Factor denominator first, then ____________________________. Multiply

numerators and denominators to get a common denominator, then _________ or _____________

the numerators.Solving Rational Equations

FACTOR FIRST!!! Find ________________________for all terms, and multiply ________________

AND_____________________ to get common denominator. Then, __________________________________

denominators and _________________________________ to solve. Don’t forget to

___________________________________!Graphing Rational Functions (Factor First!):Vertical Asymptotes/Holes - __________________________________________________________________Horizontal Asymptotes: Degree of Numerator Higher - __________ Degree of Denominator Higher - _______

Degree of Numerator and Denominator Equal - _____________________________x-intercepts - _______________________________ y-intercepts - __________________________________Example: f(x) = x 2 + 7x – 18

x - 2

VA –

Holes –

HA –

x-int: y-int:

Concept Questions:

1. Why can we “cancel out” common factors in the numerator and denominator, and why is “cancel out” not completely accurate?

2. Why do vertical asymptotes and holes exist on a rational function graph where the denominator = 0?

Unit 8 Practice Problems1.

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4. 5. 6.

Unit 9 ReviewTrigonometric

Function AbbreviationRatio of Sides

in Right Triangle

Unit Circle Coordinate

Possible Value

x-intercepts on graph

y-intercept on graph

Cosine 0 ≤ x ≤ 1

Sine sin y 0

Tangentopposite legadjacent leg 0, 1800, 3600, …

0, Π, 2Π, …

Measuring Angles on the Coordinate Plane

Angles are measured in a circle on the coordinate plane, starting at the initial side, the ___________________,

and going to the terminal side, ________________________. The angles are measured in a

_____________________________ direction.

A 900 angle has its terminal side on the __________________, and 1800 angle has its terminal side of the

___________________, and a 2700 angle has its terminal side on the _____________________. The

angle measure keeps increasing as the terminal side continues counterclockwise, even beyond ______.

Angles can be measured in degrees or radians, with _______ radians measuring the same as 3600, or a circle.

The Unit Circle

The unit circle is a circle on the coordinate plane with its center at ________________ and a radius of ______.

The key points on the unit circle are determined by __________________ and _____________________

special right triangles, and the trig values of these angles represent the coordinates of the points.

Sine and Cosine Graphs

The trigonometric ratios are functions of the ______________ they associate with, so they can be graphed as

functions on the coordinate plane. The _________________ is the x-axis, and the ________________

is the y-axis.

The graphs are __________________, as they repeat the same pattern.

Concept Questions:

1. Why are trigonometric graphs cyclical, based on the unit circle?

2. How do right triangle trigonometric ratios relate to the coordinate plane trigonometric ratios on the unit circle?

Unit 9 Practice Problems1.

2. What is the period of the sine graph at the right?A) 2 B) 4 C) Π/2 D) Π E) 2Π

3. What is the amplitude of the sine graph at the right?A) 2 B) 4 C) Π/2 D) Π E) 2Π

4.

5. 6.