Quadratic Functions, Part 1
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1 Quadratic Functions, Part 1 A2.F.BF.A.1 Write a function that describes a relationship between two quantities. A2.F.BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. A2.F.BF.A.1b Combine standard function types using arithmetic operations. Key Vocabulary: quadratic, parabola, vertex, axis of symmetry, function, degree, leading coefficient, maximum, minimum, intercepts, solutions, zeros 1. A study shows that the daily revenue from product sales can be modeled by the equation y = 5x 2 + 4, where y equals the revenue in hundreds of dollars and x equals possible increases and decreases in price. a. Analyze the function. Is the function quadratic? How do you know? Does it open upward or downward? How do you know? Will the vertex represent a maximum or a minimum? b. Graph the function. c. Determine the coordinates of the vertex. d. Create an equation that represents the axis of symmetry. e. What is the minimum daily revenue in hundreds of dollars? 2. Three times the square of a number is 5. a. Create an equation that models this situation. b. Write the corresponding quadratic function for the situation in standard form. c. Graph the function. d. Determine the coordinates of the vertex. e. Create an equation that represents the axis of symmetry.
2 3. When an object is dropped and falls to the ground under the force of gravity its height y, in feet, x seconds after being dropped is given by y = 2x a. Analyze the function. Is the function quadratic? How do you know? Does it open upward or downward? How do you know? Will the vertex represent a maximum or a minimum? b. Determine the coordinates of the vertex AND the axis of symmetry. c. Predict the height from which it was dropped? 4. A rocket is launched with an initial upward velocity of 250 feet per second from a 3 foot launch pad. The equation h = 5t t + 3 gives the rocket s height in at any given time t. a. Analyze the function. What does the vertex represent in the context of this problem? b. Predict the maximum height of the rocket. c. Evaluate the height of the rocket after 50 seconds. 5. The sales projections for a company can be represented by a quadratic equation in the form of y = ax 2 + bx + c for which c = 3, and the axis of symmetry is x = 1 2. a. Find the equation b. What is the vertex of the function? 6. Suppose a projectile is launched from ground level. If you know the velocity with which the projectile is launched, you can find the time between launch and landing using the equation h = vt 16t 2 where v is initial velocity in ft/sec and h represents the height. a. The projectile is launched with an initial velocity of 128 ft/sec. Create an equation that describes the path of the projectile. b. Write the equation for part a in standard form. c. Find the maximum height of the projectile. d. What is the significance of this formula?
3 Quadratic Functions, Part 2 A2.F.BF.A.1 Write a function that describes a relationship between two quantities. A2.F.BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. A2.F.BF.A.1b Combine standard function types using arithmetic operations. Key Vocabulary: quadratic, parabola, vertex, axis of symmetry, function, degree, leading coefficient, maximum, minimum, intercepts, solutions, zeros Consider the following functions: 1. y = 4x 2 8x a. Graph the function using the graphing calculator and analyze it. How many solutions are there to the function? How do you know? b. Find the solutions algebraically. c. Verify your solutions using the calculator. 2. y = x 2 + 5x + 6 a. Graph the function using the graphing calculator and analyze it. How many solutions are there to the function? How do you know? b. Find the solutions algebraically. c. Verify your solutions using the calculator. 3. The square of an integer is 5 more than 4 times the integer. a. Create an equation that models this situation. b. Write the corresponding quadratic function for the situation in standard form. c. Analyze the graph. How many solutions are there to the function? How do you know? d. Determine the solutions algebraically.
4 4. The square of a positive integer is 20 less than 12 times the integer. a. Create an equation that models this situation. b. Write the corresponding quadratic function for the situation in standard form. c. Analyze the graph. How many solutions are there to the function? d. Determine the solutions algebraically 5. Suppose a projectile is launched from ground level. If you know the velocity with which the projectile is launched, you can find the time between launch and landing using the equation h = vt 16t 2 where v is initial velocity in ft/sec and h represents the height. a. The projectile is launched with an initial velocity of 64 ft/sec. Create an equation that decribes the path of the projectile. b. Write the equation for part a in standard form. c. Find the maximum height of the projectile. d. Predict long does the projectile remain in the air? e. How can you tell if your answer is reasonable? 6. Compare and Contrast two methods for finding the solutions to a quadratic equation.
5 Quadratic Functions, Part 3 A2.F.BF.A.1 Write a function that describes a relationship between two quantities. A2.F.BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. A2.F.BF.A.1b Combine standard function types using arithmetic operations. Key Vocabulary: quadratic, parabola, vertex, axis of symmetry, function, degree, leading coefficient, maximum, minimum, intercepts, solutions, zeros Vertex form: y = a(x h) 2 + k 1. Analyze the graph. a. Determine the vertex. b. Determine the x-intercepts. c. Determine the degree of the polynomial. d. State the equation for the axis of symmetry. e. Create an equation for the graph in vertex form. 2. Analyze the graph. a. Determine the vertex. b. Determine the x-intercepts. c. State the equation for the axis of symmetry. d. Create an equation for the graph in vertex form. e. Write the equation in standard form.
6 3. Graph the function y = (x + 1) 2 1 a. What is the vertex? b. Determine the zeros of the function. c. Determine the y-intercept. d. Write the equation in standard form. 4. Graph the function y = (x + 4) a. What is the vertex? b. Determine the zeros of the function. c. Determine the y-intercept. d. Write the equation in standard form. 5. A firework s height h meters from the ground is given by h = 1.5t t, where t is the number of seconds after the firework has been lit. a. How many seconds have passed since the firework was lit when the firework explodes if it explodes at the maximum height of its path? b. What is the height of the firework when it explodes? c. How long is the firework in the air? 6. The path of an arrow shot from an archer s bow at a target 30 feet away can be modeled by h = 6t 1.5t 2 where h represents the height of the arrow after t seconds. a. Predict when the arrow will hit the target. b. Predict the maximum height of the arrow if the arrow is initially shot by Carson who is 5 feet tall. c. At what time will the maximum height occur?
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