Advanced Algebra Unit Review AK PDF

Advanced Algebra Maryanski Name: ______________________________ Hour:___ Unit 6 Final Exam Review (Shapes of Algebra) Review what you know about the function families: Linear: Linear functions have a constant rate of change, meaning that they are increasing or decreasin...

Advanced Algebra Maryanski Name: ______________________________ Hour:___ Unit 6 Final Exam Review (Shapes of Algebra) Review what you know about the function families: Linear: Linear functions have a constant rate of change, meaning that they are increasing or decreasing by the same number. The graph of a linear function is a straight line. Forms of Linear Equations: Slope intercept form: y = mx + b, where m is the slope and b is the y-intercept Point slope form: (y y1) = m(x x1), where m is the slope and (x1, y1) is a point on the line. Standard form: Ax + By = C Vertical Line: x = a Horizontal Line: y = b To find the slope of a line it is the change in x over the change in y. Use the slope formula: Absolute Value: Create a V shaped graph. Absolute value measures the distance from zero. Vertex form: (h, k) is the vertex, a is the rate of change one side of the vertex is positive and one side of the vertex is negative. Quadratic: Creates a U shaped graph (parabola). Standard Form: y = ax2 + bx + c Factored Form: y = a(x - #)(x - #) Vertex Form: y = a(x h)2 + k (h, k) is the vertex Exponential: J shaped curve on graph Equation Form: y= a(b)x h + k b is the constant ratio, h moves the graph left/right, k moves the graph up and down. Determine what function family the graph represents Determine what function family each equation represents Determine what function family each table represents. Explain how you know. Write the equation for the given relation Advanced Algebra Maryanski Name: ______________________________ Hour:___ Unit 7 Final Exam Review (Piecewise Functions) 1. State the Domain and Range a. b. c. d. e. f. 2. Use the piecewise function to evaluate the following a. b. c. f(-1): f(2): f(2): f(-3): f(-3): f(0): f(0): f(1): f(-4): f(-1): f(2): f(3): f(-2): f(5): f(-2): f(5): d. e. f. f(-5): f(2): f(-1): f(2): f(-1): f(2): f(0): f(1): f(0): f(1): f(0): f(1): f(-2): f(5): f(-2): f(5): f(-2): f(5): 3. Match the appropriate graph with its equation below. a. b. c. d. e. f. 4. Graph the following absolute value functions Advanced Algebra Maryanski Name: ______________________________ Hour:___ Unit 8 Final Exam Review (Polynomials) Recall: Area of Rectangle = Length x Width Perimeter: The sum of the outside 1. Perform the indicated operation if possible. Make sure the polynomial is in standard form. a. (x + 3)2 b. (2x 6)2 c. (6x2 + 8x 7) (3x2 3x + 4) d. 5x2(4x + 3) e. (x + 2)(x 3)(x + 4) f. (3x3 + 4x2 2x) + (3x3 4x + 4) g. 3x2 + 5x 7 (4x2 + 7x 8) h. 2x8 + 5x(x +4) i. 9x2 5x(3x + 2) 18 j. 7x2(2x 7) + 3x2(6x + 2) k. 5x4 (3x + 2) + 8x2 l. (3x2 + 7x)(2x 6)(4x + 3) m. (3x3 + 2x2 + x)(5x 3) n. (5x + 2)(4x2 + x) + 3x2(x 2) o. (3x2 6x) (5x2 2x + x) 2. Write an expression that represents the area of the given figure. a. 4x 3x 7x - 4 1 6x + 9 2x + 5 3. Write an expression that represents the perimeter of the given figure. a. b. 2x2 + 2x 2x2 2x2 2x2 + 4x 4. Factor out the greatest common factor a. 6x9 + 3x2 + 9 b. 4x2 + 12x + 24 c. 5x8 + 2x6 - 3x d. 21x2 + 9x e. 18x4 36x2 + 63x f. 15x3 + 5x2 25 5. Classify the following polynomial expressions a. 5x2 - 3x + 2 b. 5x + 2 c. 3x3 + 2x2 + 3x + 4 d. 8x4 e. 5x5 + 3x f. 8x8 + 6x2 + 7x - 2 Advanced Algebra Maryanski Name: ______________________________ Hour:___ Unit 9 Final Exam Review (Quadratic Functions) Determine the axis of symmetry a. (1, 45) & (7, 45) b. (-5, 14) & (-1, 14) c. (-4, 2) & (4, 2) d. (-2, -37) & (4, -37) e. f. 2. Write the Quadratic equation given the table a. b. a. b. 3. Evaluate the give expression a. f(x) = 5(x+3)2 + 1 a. g(x) = -4(x 2)2 2 b. h(x) = ½x2 10 c. b(x) = (x 2)2 + 3 f(0) = g(0) = h(0) = b(-3) = f(3) = g(3) = h(3) = b(3)= f(-4) = g(-4) = h(-4) = b(-2) = 4. Describe the transformations from the parent function a. f(x) = 5(x+3)2 + 1 b. g(x) = -4(x 2)2 2 c. h(x) = ½x2 10  d. c(x) = 5x2 e. d(x) = x2 + 4 f. r(x) = 7(x 2)2 + 8 5. Interrupting the graph a. projected from a firework. The path of the star can be modeled by the function h = -30(t 2)2 + 720, where h is the height in feet and t is the time in seconds. The graph below also models the path. Answer the following questions. a. How long will it take for the star to reach its maximum height? b. What is the maximum height? c. How high off the ground was the star shot from? d. How long until it reaches the ground? e. At what time was the star 300ft in the air? 6. Sketch the graph then fill in the blanks. a. y = x2 7x + 10 Sketch Vertex: _____________ y-intercept: _________________ Domain: ____________ Increasing Interval: __________ Range: _____________ decreasing Interval: _________ x-intercept(s): _______________________ b. y = -4(x + 2)(x 6) Sketch Vertex: _____________ y-intercept: _________________ Domain: ____________ Increasing Interval: __________ Range: _____________ decreasing Interval: _________ x-intercept(s): _______________________ c. y = - ½(x + 3)2 6 Sketch Vertex: _____________ y-intercept: _________________ Domain: ____________ Increasing Interval: __________ Range: _____________ decreasing Interval: _________ x-intercept(s): _______________________ 7. Converting between representations Equation Table Graph -4 17 -3 2 -2 -3 -1 2 0 17 1 42 Advanced Algebra Maryanski Name: ______________________________ Hour: __ Unit 10 Final Exam Review (Quadratic Equations) 1. Simplifying Radicals Product Property of Square Roots Quotient Property of Square Roots , where a, b > 0 , where a > 0 and b > 0 Rationalizing the denominator: When a radical is in the denominator of a fraction you can multiply the fraction by an appropriate form of 1 to eliminate the radical from the denominator. Simplify the following a. c. d. e. b. f. g. h. i. k. l. j. 2. Solving Quadratics Graphing Inverse Operations Factoring Quadratic Formula - Write the equation in - Isolate the quadratic - Write the equation in standard form. term (x)2 using standard form and = - Graph the function. SADMEP. 0 - Find the x-intercepts, - Take the square root - Factor completely if any of both sides (Factor our an GCF - Remember the 2 first) Solutions - Apply zero product property - Solve Solve by Graphing a. x2 = 2x + 4 b. x2 8x = - 16 Solve by any method: Leave answers exact (in simplified radical form) c. x2 18x 40 = 0 d. 16x2 = 56x e. (x + 5)2 = 36 f. 3x2 + 5x = 2 g. x2 144 =0 h. 5x2 + 2 = -7x i. x2 + 3 = 4x j. 3x2 + 7 = -6x k. 2x2 + 6x + 3 = 0 l. x2 = 2x 5 m. x(x 2) = 35 n. 3(x 2)2 18 = 26 3. Quadratics in Context a. Derek launched a model rocket directly upward at a speed of 40 feet per second from a rooftop 80 feet above ground. The function f(x) = 16x2 + 40x + 80 models the relationship between elapsed time and height of the rocket. How many seconds will it take for the rocket to be at the same height as when it was launched? b. An Internet company has found that its profit, P(x), is modeled by the function P(x) = 1.2x2 + 60x + 300 where x represents the amount of money spent in dollars on advertising. What does an x- intercept mean in the context of the problem? What are the x-intercepts in this problem and what do they represent (specific to this scenario)? c. A water balloon is catapulted into the air so that its height h, in meters, after t seconds is h = -4.9t2 + 27t + 2.4. How high is the balloon after 1 second? When will the balloon burst as it hits the ground? d. A ball is thrown up in the air. The path of the ball can be modeled by the equation h = -16t2 + 48t + 4, where h is the height in feet after t seconds. What height will the ball be when 2 seconds has passed? How long will it take the ball the reach the ground? Advanced Algebra Maryanski Name: ______________________________ Hour: __ Unit 11 Final Exam Review (Exponential Functions) 1. Properties of Exponents Product of Powers Quotient of Powers Power of a Power Power of a Product Power of a Quotient Property Property Property Property Property Definition of zero exponents Definition of Negative Exponents Simplify. Your answer should only contain positive exponents a. b. c. d. e. f. g. h. i. j. k. l. m. n. o. 2. Writing Exponential Functions. Write the equation of the table given. a. b. c. d. 3. Exponential Growth Vs. Decay. Graph each given below, fill in the table, asymptote, domain, range, state if growth decay or neither, and then compare each graph to its own parent function. a. f(x) = 3 2x + 1 2 Growth Decay Neither Asymptote: ___________ Domain: ________________ Range: _______________ Parent Function: ____________________ Transformations: b. f(x) = ½ 2x 1+ 6 Growth Decay Neither Asymptote: ___________ Domain: ________________ Range: _______________ Parent Function: ____________________ Transformations: c. f(x) = -2 4x 6 Growth Decay Neither Asymptote: ___________ Domain: ________________ Range: _______________ Parent Function: ____________________ Transformations: Advanced Algebra Maryanski Name: ______________________________ Hour: __ Unit 12 Final Exam Review (Data and Statistics) 1. Box and Whisker Plots a. Twenty middle school students ran the mile and their times (minutes) were recorded. Use the data to make a box and whisker plot. 8, 7.5, 6, 5.75, 10.25, 13.75, 6.5, 8.75, 9, 8, 9.25, 10, 8.25, 8.25, 9.25, 11, 6.75, 7, 9.5, 8.5 b. The following box and whisker plots show surfboard prices for two surf shops. a. Compare the medians for the surf shops. c. Which surf shop has prices that are more Which one is greater and by how much? spread out? Justify your answer. b. What percentage of surf board prices fall d. What percentage of surf board prices fall between $400 and $600 dollars for shop B? between $400 and $600 dollars for shop A? c. The following box and whisker plot represents the scores earned on a math test. a. What is the median score? d. Which statement is NOT true about the box and whisker plot show? (1) 75 represents the mean score b. What score represents the lower quartile (2) 100 represents the maximum score (Q1)? (3) 85 represents the 3rd quartile (4) 55 represents the minimum score c. What is the range of the scores? 2. Histograms a. The graph below shoes the distribution of scores of 30 students on a math test. a. Complete the frequency table from the histogram. b. What percent of students got above a 70%? c. Can you find the mean, median, mode, or range from the histogram? Explain. b. The scores on a science test were 70, 55, 61, 80, 85, 72, 65, 40, 74, 68, and 84. Complete the frequency table then construct a histogram for this data set. a. What percent of students scored 50% or better? b. Is the histogram skewed? Explain. 3. Dot plots a. The dot plot below represents the number of siblings that students have. a. Calculate the following: Mean: Median: Mode: Range: b. What percent of students have 3 or more siblings? c. How many students have 2 siblings or less?