Algebra 2 Unit 4 Part 2 Homework Key PDF

# ALGEBRA 2 UNIT 4 PART 2 ASSIGNMENTS ## UNIT 4 DAY 8 HOMEWORK: Graphs and Transformations of Quadratic Functions Identify the transformations from the parent function f(x) = x². Then graph the function. | Function | Transformations | |---|---| | f(x) = (x + 4)² - 2 | Translation left 4 units and...

# ALGEBRA 2 UNIT 4 PART 2 ASSIGNMENTS ## UNIT 4 DAY 8 HOMEWORK: Graphs and Transformations of Quadratic Functions Identify the transformations from the parent function f(x) = x². Then graph the function. | Function | Transformations | |---|---| | f(x) = (x + 4)² - 2 | Translation left 4 units and down 2 units | | f(x) = (x - 3)² + 4 | Reflection about the x-axis. Translation right 3 units and up 4 units | | f(x) = (x - 3)²/2 | Vertical compression by a factor of 1/2. Translation right 3 units | | f(x) = -2x² + 2 | Reflection about the x-axis, vertical stretch by a factor of 2, translation up 2 units | ## Write the equation of the function described. 1. A quadratic function reflected over the x-axis, translated 7 units left, and vertically stretched by a factor of 5. y = -5(x + 7)² 2. A quadratic function vertically compressed by a factor of 1/4 and shifted 8 units up. y = (x - 1)² /4 + 8 ## Write the equation of the function graphed. 1. The graph is a parabola opening downward with vertex at (1, 1). y = -(x - 1)² + 1 2. The graph is a parabola opening upward with vertex at (-2, 2). y = a(x + 2)² + 2 # UNIT 4 DAY 9 HOMEWORK: Quadratic Functions in Standard and Vertex Form, Analyzing Quadratic Functions ## Find the axis of symmetry, the coordinates of the vertex, and then graph the function. 1. y = (x + 3)² - 1 - Transformations: Vertical compression by a factor of 1/3. Translation left 3 units and down 1 unit - Axis of Symmetry: x = -3 - Vertex: (-3, -1) 2. y = x² + 4x + 4 (Hint: use x = -b/2a to find the axis of symmetry) - Axis of Symmetry: x = -2 - Vertex: (-2, 0) - Zeros: x = -2 - y-intercept: y(0) = 4 ## Given the function described, write the equation of the function in vertex form, and then state the vertex, and the axis of symmetry. 1. A quadratic function vertically compressed by a factor of 1/2, and reflected over the x-axis. - Equation: y = -1/2x² - Vertex: (0, 0) - Axis of Symmetry: x = 0 2. A quadratic function shifted to the right 4 and shifted up 7. - Equation: y = (x - 4)² + 7 - Vertex: (4, 7) - Axis of Symmetry: x = 4 3. A quadratic function reflected over the x-axis, vertically stretched by a factor of 3, shifted left 2 and down 10. - Equation: y = -3(x + 2)² - 10 - Vertex: (-2, -10) - Axis of Symmetry: x = -2 # UNIT 4 DAY 10 HOMEWORK: QUIZ REVIEW ## Solve the quadratic inequalities. Write answers in interval notation. 1. x² - 11x ≥ -28 - (-∞, 4] ∪ [7, ∞) 2. x² + x - 8 < -6 - (-2, 1) 3. 6x² + 11x - 10 > 0 - (-∞, -5/2) ∪ (2/3, ∞) 4. 8x² - 12 ≤ 2x² - 7x - (-∞, -3/2] ∪ [2, ∞) ## Identify the transformations from the parent function f(x) = x². 1. g(x) = -1/2(x - 8)² + 7 - Reflection about the x-axis, vertical stretch by a factor of 1 / 2, Translation right 8 units and up 7 units ## Write the equation of a quadratic function vertically stretched by a factor of 4, shifted left 3 units, and down 5 units. - y = 4(x + 3)² - 5 # UNIT 4 DAY 11 HOMEWORK: Solving Systems of Equations Graphically ## Solve each system by graphing (hand or calculator). Write your solutions as ordered pairs. 1. y = (x + 1)² - 4 2x - y = -1 - Solutions: (-2, -3) and (2, 5) 2. y = -(x - 1)² - 2 y = (x + 2)² - 7 - Solutions: (-2, 0) and (0, 0) 3. y = -x² + 3x + 2 y = 3x + 2 - Solutions: (0, 2) 4. x - y = 3 y = x² - 2x + 1 - Solutions: (-1, 8), (4, 3) 5. x - y = 3 y = x² - 2x + 1 - Solutions: (3 - √ 7 / 2, 3 - √ 7 / 2 ) and (3 + √ 7 / 2, 3 + √ 7 / 2) 6. x² - 4x + 3 = -2x + 6 x - y = 3 - Solutions: (-1, 8), (3, 0) # UNIT 4 DAY 12 HOMEWORK: Solving Systems of Equations Algebraically ## Solve each system using either substitution or elimination. Write answers as ordered pairs. 1. x + 3y = 7 2x - 4y = 24 - (10, -1) 2. y = x² + 5x - 2 y = 3x - 2 - (-2, -8) and (0, -2) 3. 3x + 2y = 10 6x + 4y = 15 - No Solutions 4. 5x - 2y = -19 2x + 3y = 0 - (-3, 2) 5. y = -x² - 3x - 2 y = x² + 3x + 2 - (-2, 0) and (1, 0) 6. 3x + 2y = 10 9x + 6y = 30 - All solutions of the equation 3x + 2y = 10 7. y = x² - 4x + 7 y = -2x² + 20x - 41 - (4, 7) 8. y = -x² - 4x + 2 y = x² + 8x + 12 - (-5, -3) and (-1, 5) # UNIT 4 DAY 13 HOMEWORK: Lines and Curves of Best Fit 1. Find the equation of the linear regression model that best fits the data. Round all numbers to two significant digits after the decimal place. - X | y - -3 | 7 - -2 | 4 - 0 | 1 - 5 | -9 - 6 | -11 - 11 | -21 - 20 | -39 - Y ≈ -1.98x + 0.76 2. Find the equation of the quadratic regression model that best fits the data. - X | y - -2 | 1 - -1 | 4 - 0 | 9 - 2 | 25 - 3 | 36 - 4 | 49 - Y = x² + 6x + 9 3. What type of correlation most closely fits the scatterplot below? - a.) What equation most closely fits the scatterplot? - Y ≈ -0.28x² + 0.23x + 0.55 - b.) What is the name of the shape of the graph? - Graph of a quadratic function (Parabola) 4. The table below shows the number of hours studied and math test score percentages. - Number of hours | Score (%) - 8 | 75 - 5 | 62 - 12 | 80 - 10 | 85 - 2 | 35 - 9 | 70 - 11 | 82 - 14 | 95 - Y ≈ 4.47x + 33.31 - How many hours would a student need to study to score a 100%? - Approximately 14.30 hours # UNIT 4 DAY 14 HOMEWORK: Real-Life Applications with Quadratics 1. A baseball coach records the height of at every second of a ball thrown in the air. Some of the data appears in the table below. - Time (s) | Height (ft) - 0 | 0 - 1 | 64 - 3 | 96 - h = -16t² + 80t - What is the height of the ball at 2.5 seconds? - 80 ft 2. While playing basketball this weekend, Frank shot an air-ball. The time, t, in seconds, and the height, h, in feet, of the ball is modeled by h(t) = -16t² + 32t + 8. - a. How long will it take the ball to strike the ground? - Approximately 2.22 seconds - b. What is the maximum height of the ball? - 24 feet 3. A rectangular window has a length that is four more than twice the width. The area of the window is 48 square feet. What is the length of the window? - The length is 12 feet. 4. Jason dives off a cliff into the ocean. His height as a function of time can be modeled by the function h(t) = -16t² + 16t + 480, where t is the time in seconds and h is the height in feet. - a. How long did it take Jason to reach his maximum height? - 1/2 second - b. What was the highest point that Jason reached? - 484 feet - c. Jason hit the water after how many seconds? - 6 seconds 5. The length of a rectangular carpet is six more than twice the width. If the area of the carpet is 80 square feet, what are the dimensions of the carpet? - The dimensions are 5 feet by 16 feet. 6. The table at the right shows average retail gasoline prices. - a. Find the equation of the curve of best fit using 1976 as year 0, 1986 as year 10, 1996 as year 20, and 2005 as year 29. - Y ≈ 0.11x² + 1.87x + 61.83 - b. Use the model to estimate the average retail gasoline price in 2015. - Approximately 300.17 cents # UNIT 4 PART 2 TEST REVIEW ## Graph f(x) = (x + 3)² - 3 - Transformations: Translation left 3 units and down 3 units. ## Graph f(x) = 1/2 (x - 4)² + 2. - Transformations: Reflection about the x-axis, vertical compression by a factor of 1/2, Translation right 4 units, and up 2 units. ## Identify the transformations from the parent function of a quadratic. Then write the equation of the function. - Transformations: Vertical stretch by a factor of 2, translation right 3 units and down 5 units. - y = 2(x - 3)² - 5 ## Write the equation of a quadratic function reflected over the x-axis and shifted 7 units up. - y = -x² + 7 ## Write the equation of a quadratic function vertically compressed by a factor of 1/5 and shifted 4 units to the left. - y = 1/5(x + 4)² ## Given a quadratic function in standard form, find the axis of symmetry, the vertex, and the y-intercept. - y = 2x² - 4x - 9 - Axis of Symmetry: x = 1 - Vertex: (1, -11) - y-intercept: -9 ## Given a quadratic function in standard form, find the axis of symmetry, the vertex, and the y-intercept. - y = -3x² + 12x - 5 - Axis of Symmetry: x = 2. - Vertex: (2, 7) - y-intercept: -5 ## Given f(x) = (x - 2)² - 1. Graph the function. - Domain: (-∞, ∞) - Range: [-1, ∞) - Left End Beh: As x → -∞, f(x) → ∞ - Right End Beh: As x → ∞, f(x) → ∞ - Zero(s): 1 and 3 - Y-intercept: 3 - Increasing Interval: (2, ∞) - Decreasing Interval: (-∞, 2) - Vertex: (2, -1) - Axis of Symmetry: x = 2 ## Given f(x) = -2(x + 1)² + 2. Graph the function. - Domain: (-∞, ∞) - Range: (-∞, 2] - Left End Beh: As x → -∞, f(x) → -∞ - Right End Beh: As x → ∞, f(x) → -∞ - Zero(s): - 2 and 0 - Y-intercept: 0 - Increasing Interval: (-∞, -1) - Decreasing Interval: (-1, ∞) - Vertex: (-1, 2) - Axis of Symmetry: x = -1 ## Solve the following quadratic inequalities and write your answer in interval notation. 1. 6x² + 14x > -8 - (-∞, -1) ∪ (-4/3, ∞) 2. 8x + 9 ≤ - x² - [-4 - √ 7, 4 + √ 7 ] ## Using transformations, graph the following functions and find all real solutions. Write any solutions as ordered pairs. If there are no real solutions, write “none”. 1. y = (x + 1)² y = -2x - 4 - (-3, 2) and (-1, -2) 2. y = 2(x + 5)² - 1 y = (x + 5)² + 2 - (-6, 1) and (-4, 1) ## Solve each system using substitution or elimination. Write any real solutions as ordered pairs. 1. 3x - 5y = -2 2x + 6y = 8 - (1, 1) 2. 3x - y = 6 -6x + 2y = 8 - No Solutions 3. 4x + 8y = 3 16y = -8x + 6 - All solutions to the equation 4x + 8y = 3 4. y = x² + 5x + 6 y = -x² + 5x + 6 - (0, 6) and (4, 10) # Graphing Calculator Permitted ## Find the linear equation, in ax+b form, that best fits the data. Round to the hundredths. - X - 3 - 4 - 5 - 7 - 8 - 9 - 10 - Y - 5 - 7 - 9 - 10 - 10 - 11 - 13 - Y ≈ 0.96x - 2.36 - What would the x-vale be when y = 20? - Approximately 16.88 ## The table shows the height of a javelin as it is thrown and travels across a horizontal distance. Find the quadratic model to represent the path of the javelin. Round to the nearest hundredth or 2 non-zero digits. - Distance (m) | Height (m) - 5 | 2 - 18 | 5 - 33 | 8 - 55 | 6 - 68 | 4 - 74 | 3 - Y ≈ -0.0043x² + 0.35x + 0.37 - What would be the predicted height at a distance of 60m? - Approximately 5.85 meters ## The function h = -0.1t² + 0.9t models the height, h, in feet, of a soccer ball, t seconds, after it has been kicked. How high did the soccer ball reach? How long did it take for the soccer ball to reach the ground? - How high: 20.25 feet - How long did it take to reach the ground: 9 seconds ## If a toy rocket is launched vertically upward from ground level with an initial velocity of 128 ft/sec, then its height, h, after t seconds, is given by the equation h(t) = -16t² + 128t. - a. How long will take the rocket to hit the ground? - 8 seconds - b. How long will it take the rocket to reach its maximum height? - 4 seconds - c. What is the maximum height of the rocket? - 256 feet ## The length of picture frame is one less than twice the width. If the area of the frame is 15 in², find the dimensions of the frame. - The dimensions are 3 inches by 5 inches. ## Use the system of equations to answer the following questions. 1. x - y = -6 x² + 3x = y + 2 - a. Identify the types of equations: - Linear and quadratic - b. Number of real solutions: - 1 - c. Write any real solutions as ordered pairs: - (-2, 4) 2. x² + x + y = 12 y = x² + 7x + 12 - a. Identify the types of equations: - Quadratic and quadratic - b. Number of real solutions: - 1 - c. Write any real solutions as ordered pairs: - (-4, 0) and (0, 12)