Quadratics and Their Graphs

Transcription

1 Quadratics and Their Graphs Graph each quadratic equation to determine its vertex and x-intercepts. Determine if the vertex is a maximum or minimum value. y = 0.3x + 3x 1 vertex maximum or minimum (circle one) x-intercepts solutions when y = 0 What is the value for y when x = 10? y 1 ( x 4) vertex maximum or minimum (circle one) x-intercepts solutions when y = 0 What is the value for y when x = 6?

2 Quadratic Match Cut apart and match the Representation Cards to each context. Answer the questions posed for each context and write your answer in the answer column. Identify the solution strategy that you used for the problem. Your choices include factoring, completing the square, the quadratic formula, using a graph, and using a table. The Clue Box contains hints for writing the functions and an answer bank. The answer bank contains all possible answers, and some may be used more than once. Answers are rounded to the nearest tenth. Clue Box Hint: When an object is dropped, its movement can be modeled by the function f(x) = 16x + h 0, where x is elapsed time in seconds and h 0 is the original height in feet. Hint: When an object is thrown, its movement can be modeled by the function f(x) = 16x + V 0 x + h 0, where x is elapsed time in seconds, V 0 represents the object s initial velocity in feet per second, and h 0 is the original height in feet. If the object is thrown upward, then V 0 will be positive. If the object is thrown downward, then V 0 will be negative. Possible Answers Context Representation Cards Questions Answers Solution Strategy Alex dropped a water balloon from 3 ft above the ground and hit Michael, who is 5 feet tall, on top of the head. x represents elapsed time in seconds. How many seconds was the water balloon in the air before it hit Michael? How many seconds had passed when the water balloon was 8 feet above ground? The accountant for Fun Splash Jet Ski Rentals uses the function f(x) = 0.5x 14x to represent the monthly operating costs, where x represents the number of jet skis rented in a month. If the monthly operating costs totaled $304, how many jet skis were rented that month? According to the function, what is the lowest possible monthly operating cost?

3 Context Representation Cards Questions Answers Solution Strategy A ball is kicked from the ground into the air at a velocity of 5 ft/sec, where x represents the elapsed time in seconds. At 1. seconds, how far is the ball from the ground? When is the first time the ball is 8.5 feet from the ground? A recording company uses the function f(x) = 80x x to model the number of downloads of a new single recording, where x represents the elapsed number of days. Which day is the first time there were 78,000 downloads? Which day had the most downloads? Madison was in the bleachers and threw a volleyball downward with an initial velocity of 10 ft/sec from a height of 40 feet, where x represents the elapsed time in seconds. How long did it take for the volleyball to hit the floor? When was the ball 31 feet above the ground?

4 Two sets are provided. Representation Cards x f(x) f( x) 16x 3 f( x) 16x 5x x f(x) 10 56, , , ,000 x f(x) f( x) 16x 3 f( x) 16x 5x x f(x) 10 56, , , ,000

5 How Do I... * Solve a quadratic equation by factoring? Write the quadratic equation in standard form. 7x 4x 11 0 If the coefficient of x is greater than one, divide both sides of the equation by that value. Write the quadratic expression as a product of two binomials. 7x 4x 11 0 x 7 7 6x 18 0 ( x 8)( x ) 0 Solve each binomial for the indicated variable. x 8 0 or x 0 x 8 or x Solve a quadratic equation by using the quadratic formula? Write the quadratic equation in standard form. 7x 4x 11 0 Identify the values for a, b, and c. a 7, b 4, c 11 Substitute the values for a, b, and c into the quadratic formula. x 4 4 4(7)( 11) (7) Solve for x. x or x 8

6 Solve a quadratic equation by completing the square? Divide each side by the coefficient of x. 7x 4x x 4x Write the left side of the equation in x + bx form. x 6x 16 0 x 6x 16 6 Add equation. 3 9 to each side of the Rewrite the left side of the equation as a binomial square. x 6x ( x 3) 5 Take the square root of each side. ( x 3) 5 x 3 5 Solve for x. x 35 or x 35 8

7 Solving Quadratics I should consider solving a quadratic equation using factoring when... I should consider solving a quadratic equation by completing the square when... I should consider solving a quadratic question by applying the quadratic formula when... Choose a method to solve the following. Provide an explanation why you chose the particular method. Marcos needs to use a function f(x) to find the area of particular rectangles whose lengths are 8 cm greater than the widths, x. If the area of the rectangle is 09 cm, what is the length? If the width of the rectangle is 13 cm, what is the area? If the width of the rectangle is 6 cm, what is the length?

8 Directions: Choose one of the Quadratic Situation Cards from your teacher. With your partner, create a small poster to represent your work. Your poster should include some kind of visual, and your solution can be either a graphic or algebraic version.

9 Quadratic Situation Cards The neighborhood park wants to add a gravel walkway around the memorial rose garden. The walkway will be the same width around the perimeter and will cover 168 square feet. If the rose garden is 10 feet by 1 feet, how wide should the construction crew make the walkway? The community center wants to add an enclosed patio area somewhere along its 80-foot west side. The patio will be enclosed by 1400 feet of fencing on 3 sides. What are the dimensions of the patio? The height of a triangle is 4 cm less than four times its base. If the area of the triangle is 11 cm, write an equation and find the dimensions of the triangle in centimeters. A squirrel drops an acorn from a tree limb 5 feet above the ground. Assuming the acorn does not hit anything on its way down, how many seconds will it take the acorn to hit the ground? An Olympic platform diver dives from a platform that is 3 feet tall. How many seconds does he have in the air to complete his series of turns and twists? An object is launched upward with an initial velocity of 48 feet per second from 64 feet. Write a function to model this description. What is the maximum height of the object? How many seconds for the object to reach the maximum height? How many seconds for the object to hit the ground? A local florist has modeled her monthly operating costs for making large floral arrangements by the function y = 0.5x 0x + 475, where y is the total cost in dollars and x is the number of large floral arrangements created. What is her minimum cost? How many large arrangements can she create each month and stay under her minimum cost? A music store sells beginner guitar sets. The manager uses the equation y = 10x + 400x + 50 to model his monthly revenue, y. How many guitar sets, x, should he sell to earn the maximum revenue? The local community theater sells about 180 tickets to each play and charges $0 per ticket. Their expenses each week are $1600. The financial manager noticed that if they reduce the price of the ticket by $1, they should sell about 1 more tickets. The manager uses the equation y = 1x + 60x to represent the monthly profit, y, for the theater, where x is the number of $1 reductions in ticket prices. How much should the manager recommend that the theater reduce the ticket price in order to maximize their profits?

10 * Answer each of the following questions and create a poster for one of the problems to display your work. Your poster should include some kind of visual, and your solution can be either a graphic or algebraic version. I. Triangle a. Label the picture with the given information. b. What is the formula for the area of a triangle? c. Write a quadratic equation for this situation. d. Choose a method to solve your equation. You may use a graph, the quadratic formula, factoring, completing the square, or finding the square roots. e. Is your answer reasonable? How can you verify your solution?

11 II. Clumsy Squirrel a. Is the acorn being thrown or dropped? b. What information in the clue box from the previous activity could help you? c. Write a quadratic equation for this situation. d. Choose a method to solve your equation. You may use a graph, the quadratic formula, factoring, completing the square, or finding the square roots. e. Is your answer reasonable? How can you verify your solution?

12 Evaluate: 1 A cattle rancher needs to build a temporary holding pen for some extra cattle she has purchased. She would like to build a rectangular region using one side of her barn and 00 feet of fencing. What is the maximum area she can enclose? Record and bubble your answer in the grid provided. Holding Pen Barn During football tryouts, the quarterback coach has the players throw footballs for distance to various targets on the field. On your turn, your football left your hand 5 feet above the ground and had an initial vertical velocity of 45 feet per second upward. About how long was your football in the air? A B C 5 sec.9 sec 1.4 sec D 0.1 sec 3 A certain rectangle has an area of 165 cm. The length of the rectangle is 4 cm greater than its width. Which equation could be used to find the width, x? A xx ( 4) 165 B x ( x 4) 165 C x(4 x ) 165 D x( x 4) The financial analyst for the Municipal Cable Company uses the equation y = 6x + 18x to represent the company s monthly revenue, where x represents the number of $0.10 increases in the movie price. They presently charge $4.00 per movie. What price for movies should the cable company charge to maximize its monthly revenue? A $4.55 B $4.95 C $5.5 D $6.08