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44767533337500Unit 1 Review49580521907500Parallelogram PropertiesRemember, 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 otherDiagonals that divide the parallelogram into ________________________Formulas to help: Slope = Distance = Midpoint = Volume FormulasVolume of a right prism (including a rectangular prism) or cylinder = Area of __________ ? ______________Volume of a cone or pyramid = Area of ___________●_____________3Volume of a sphere = 4Πr33 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 domain3. Push 2nd - Trace (Calc) - #3 minimum or #4 maximum, depending on the question asked4. Set your ________________ to the right and left of the minimum or maximum point, then get your valueConcept 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?2876550371474006667537147500Unit 1 Sample Problems1. 2. 133350281305002895600285750003. 4. 133350245745002962275231140005. 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 = 8CirclesArea 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 MeasureRadians = ______________Degrees = _________________Examples: Convert 5π4 radians to degrees.Convert 3000 to radians.Radian Measure in Circles: Arc Length = __________________ ? _______________Angle-Arc RelationshipsCentral Angle - ______________ measure of intercepted arc, vertex at the ____________Inscribed Angle - ______________ measure of intercepted arc, vertex on the _____________42862520764500Circumscribed Angle - Angle formed by _____________ lines to circle336232524574500Circumscribed Angle4391025-29527500Circle SegmentsChord - Segment connecting two points _________________ Tangent - Segment that touches a circle in ________________Secant - Line/segment that intersects a circle in __________________487680024701500281940029464000-32385031369000Secant-Tangent IntersectTwo Chords IntersectTwo Secants IntersectConceptual Questions:1. How does the Pythagorean Theorem relate to the equation of a circle?24669754679952. 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 Problems2876549120650012382512065001. 2. 123825277495003. 2876550176530004. 3790950161290006. -209550222250005. 3790950194310007.Unit 3 ReviewStatisticsExperiment – Study comparing a _____________ group and an _______________ groupObservation – Study observing the characteristics of __________ groupSimple 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 meanMargin of Error - Expected value that a sample mean could deviate from the actual ____________________Margin of Error Formula: Standard Deviationn (where n is the sample size)Bias - Unintended feelings or actions that skew dataConcept 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 Problems-180975180340001. -257175229870002. 7242171356360003. 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) 10B) 13C) 64D) 2104. 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 FunctionsInverse 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 FunctionsFor 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 FunctionsExponential functions, with a _________________ as the exponent, use an operation called ______________to determine the inverse.To convert exponents to logarithms:bx = a → logb a = x10x = a → log a = xex = 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?)3295650333375009525033337500Unit 4 Practice Problems1. 2. 95250207010003. Assuming f(x) represents the number of families as a function of the year, what is f-1 (30,100)?A) 65B) 70C) 75D) Not enough information-2857522288500Use 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 ≤ -0.5B) x ≥ -0.5C) x ≤ -6.5D) x ≥ -6.55) What is f-1 (-3)?A) -6.5B) -3C) -0.5D) 1Unit 5 ReviewExponential Functionsy = 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 ExponentTo 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 logarithm5. 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 bothsides of the equation and finding the ______________________________ using technology.Change of Base Formulalogb x = logxlogb = lnxlnbWe can evaluate log and ln, the common logarithms, using the calculator to get a decimal approximation.Graphs of Exponential Functionsy = abxa = _________________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?10477536195000Unit 5 Practice Problems1. 104775244475002. 289560020955000104775285750003. 4. 104775203199005. 6. Two population functions are graphed on the same plane to compare their growth. The first, f(x), represents one town’spopulation 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 thefunctions?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 FunctionsThe 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.Inside FunctionOutside FunctionPositive (+)Negative (–)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 Functions266700028130500Greatest 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: 266700027495500Least 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: 266700024638000Piecewise FunctionsPiecewise functions are functions with different function rules for different ________________.For example, they can be represented: f(x) = -2x, for x ≤02x, for 0<x<5x2,for x ≥5The 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 - 4Evaluate 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?3514725409575005715134290000Unit 6 Practice Problems1. 2. 152400224155003. 152400249555004. 5. For f(x) = 2x2 + 8x, g(x) = 2x, and h(x) = x + 4, what isf(x) - g(x) ? h(x)?A) 0B) 1C) 2x3 + 14x2 + 24xD) 9x2 + 36xUnit 7 ReviewKey TermsDegree - ________________________________________________________________________________Leading Coefficient - ______________________________________________________________________Solution - _______________________________________________________________________________Graphs of PolynomialsGraphs 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 -∞:Odd DegreeEven DegreePositive Leading CoefficientAs x → - ∞, y __________As x → ∞, y ____________As x → - ∞, y __________As x → ∞, y ____________Negative Leading CoefficientAs x → - ∞, y __________As x → ∞, y ___________As x → - ∞, y __________As x → ∞, y ___________Dividing PolynomialsLong Division - To divide any polynomial by another polynomial.-39878035877500Synthetic Division - Shortcut to divide a polynomial by a binomial (with leading coefficient of 1)275145513906500Remainder 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 AlgebraFundamental Theorem of Algebra - The number of solutions, real or complex, to any function is equal to its_________________.Building Polynomials from RootsIf 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 PolynomialsTo 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 samecoordinate 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 Problems476250016446500-133350164465001. 5. -133350209550002. 491490024701500-133349156210003. 6. -180974278766004. Unit 8 ReviewOperations With Rational ExpressionsRational expressions follow the same arithmetic rules as _____________________.Multiply rationals - Factor first, then ___________________________, then divide out common factorsDividing rationals - Factor first, then ______________________________, then divide out common factorsAdding or subtracting rationals - Factor denominator first, then ____________________________. Multiplynumerators and denominators to get a common denominator, then _________ or _____________the numerators.Solving Rational EquationsFACTOR 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 - __________________________________389572510287000Example: f(x) = x2 + 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?5715019050000Unit 8 Practice Problems1. 47625002393950013335043815002.3. 5191125128270002905125289560004. 152400-635005. 6. Unit 9 ReviewTrigonometric FunctionAbbreviationRatio of Sides in Right TriangleUnit Circle CoordinatePossible Valuex-intercepts on graphy-intercept on graphCosine0 ≤ x ≤ 1Sinesiny0Tangentopposite legadjacent leg0, 1800, 3600, …0, Π, 2Π, …Measuring Angles on the Coordinate PlaneAngles 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 CircleThe 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 GraphsThe trigonometric ratios are functions of the ______________ they associate with, so they can be graphed asfunctions 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 Problems-247650202565001. 4276725112395002. What is the period of the sine graph at the right?A) 2B) 4C) Π/2D) ΠE) 2Π3. What is the amplitude of the sine graph at the right?A) 2B) 4C) Π/2D) ΠE) 2Π4. 152400-381000331470526670000152400266700005. 6. 497205015811500 ................
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