Unit 7. Quadratic Applications. Math 2 Spring 2017

Transcription

1 1 Unit 7 Quadratic Applications Math 2 Spring

2 Contents Graphing Key Features of Quadratic Equations...3 Vertex Form of a Quadratic...3 Practice and Closure...6 Graphing Quadratics from Standard Form...7 Quadratic Applications...9 Practice and Closure Practice and Closure Systems of Quadratics Practice and Closure Systems of Quadratics Practice and Closure Quadratic Inequalities Practice and Closure Cartesian Island Unit 7 Review

3 Graphing Key Features of Quadratic Equations 3 A in standard form is written y = ax 2 + bx + c where a 0. A is the U-shaped graph of a quadratic. The is an imaginary line that passes through the vertex and divides the graph in half. It is represented by a. The is the lowest or highest point of a parabola. It is also known as either the or point of the parabola. Vertex Form of a Quadratic 1. In unit 2, we looked at transformations Let s explore TI Interactive! a h k Vertex Quadratic Equation: y a x h 2 k 2. Let s translate each parabola to find the vertex, then use the a value to graph! a) y x b) y x c) y x 2 4 Vertex: Vertex: Vertex: 3

4 Working backwards: You know the transformation, you have to write the equation. 4 2 d) f x x is stretched vertically by 3 and translated left 2 2 e) f x x is reflected across the x-axis and translated 3 units up Applications 1. The graph below illustrates the area of a room in a newly remodeled house. The smaller parabola was the original size, and the larger one represents the new size. Write the function of each parabola as a transformation of the function f ( x) write the function for the larger size as a transformation of the function of the original size. Function of Old Room: x 2. Next, Function of New Room: Function of Old to New: 2. An object is launched from a platform. Its height (in meters), x seconds after the launch, is modeled by: h(x) = 5(x 4) How many seconds after being launched will the object hit the ground? 3. Ming throws a stone off a bridge into a river below. The stone's height (in meters above the water), x seconds after Ming threw it, is modeled by: h(x) = 5(x 1) What is the maximum height that the stone will reach? 4

5 4. The power generated by an electrical circuit (in watts) as a function of its current c (in amperes) is modeled by P(c) = 20(c 3) Which currents will produce no power (i.e. 0 watts)? 5 Lower current: Higher current: amperes amperes 5. A certain company's main source of income is selling socks. The company's annual profit (in millions of dollars) as a function of the price of a pair of socks (in dollars) is modeled by P(x) = 3(x 5) What is the maximum profit that the company can earn? 6. Ying is a professional deep water free diver. His altitude (in meters relative to sea level), xxx seconds after diving, is modeled by: D(x) = 1 36 (x 60)2 100 How many seconds after diving will Ying reach his lowest altitude? 5

6 Practice and Closure For each of the following, identify the following: 6 a) Vertex b) Axis of Symmetry c) Max/Min d) Stretch or compression 6

7 Graphing Quadratics from Standard Form 7 Yesterday we looked at quadratics in form. Ex: 2 y 3 x 1 2 This is convenient because we can easily identify the vertex: stretch. However, sometimes quadratics are written in form: and the To review, let s convert 2 y 3 x 1 2 into standard form. From this form, we still want to be able to find the vertex (and other things) Graphing Quadratics by Hand Find the vertex, axis of symmetry and the y-intercept. Graph each quadratic function. 1. y = x 2 + 2x + 1 Axis of Symmetry: x y Vertex: Y-intercept: 7 2. y = x 2 4x + 5

8 8 Axis of Symmetry: x y Vertex: Y-intercept: 3. y = 2x 2 8x + 1 Axis of Symmetry: x y Vertex: Y-intercept: 8

9 9 4. y = x 2 + x + 4 Axis of Symmetry: x y Vertex: Y-intercept: Quadratic Applications 5. A ball is thrown straight up with an initial velocity of 56 feet per second. The height of the ball t seconds after it is thrown is given by the formula h(t) = 56t 16t 2. a. What is the height of the ball after 1 second? b. What is its maximum height? c. After how many seconds will it return to the ground? 9

10 6. An object is thrown upward into the air with an initial velocity of 128 feet per second. The formula h(t) = 128t 16t 2 gives its height above the ground after t seconds. 10 a. What is the height after 2 seconds? b. What is the maximum height reached? c. For how many seconds will the object be in the air? You Try: A baseball is projected upward from the top of a 448 foot tall building with an initial velocity of 48 feet per second. The distance, s, of the baseball from the ground at any time t, in seconds, is given by the equation s(t) = 16t t a. Find the time it takes for the baseball to strike the ground. b. What is the baseball s maximum height? 7. A rocket is shot upward such that its height in feet, h, is given as h(t) = 62t 5t 2, where t is the number of seconds since liftoff.. a. When is the ball going to be 120 feet in the air? b. How long will the ball me more than 120 feet in the air? 8. What is the largest area that can be enclosed with 400 feet of fencing? What are the dimensions of the rectangle? 9. Lorenzo has 48 feet of fencing to make a rectangular dog pen. If a house were used for one side of the pen, what would be the length and width for the maximum area? 10

11 Practice and Closure Graph the following quadratic functions. Identify the key characteristics y x 4x 1 a. Direction of Opening: b. Axis of Symmetry: c. Vertex: d. Minimum or Maximum? e. y-intercept: 2 2. y x 4x 10 a. Direction of Opening: b. Axis of Symmetry: c. Vertex: d. Minimum or Maximum? e. y-intercept: 3. y 4x 2 16 a. Direction of Opening: b. Axis of Symmetry: c. Vertex: d. Minimum or Maximum? e. y-intercept: 11

12 12 4. y 3x 2 24x 80 a. Direction of Opening: b. Axis of Symmetry: c. Vertex: d. Minimum or Maximum? e. y-intercept: y 15 6x x 5. a. Direction of Opening: b. Axis of Symmetry: c. Vertex: d. Minimum or Maximum? e. y-intercept: 2 6. y 5 16x 2x 2 a. Direction of Opening: b. Axis of Symmetry: c. Vertex: d. Minimum or Maximum? e. y-intercept: 12

13 Practice and Closure 13 Solve the following word problems: 1. The revenue, R, made selling phones at price, p, can be modeled by R = 5 2 p p a) What price will maximize the companies revenue? b) What is the maximum revenue? 2. The revenue, R, made selling widgets at price, p, can be modeled by R = -5p p a) What price will maximize the companies revenue? b) What is the maximum revenue? 3. The available power, P, is a function of the amount of current flowing in amperes P = 120A 20A 2 a) How many amperes will produce the maximum power? b) What is the maximum power? 4. The profit, P, a company makes depends on the ticket price, t, they charge. P = 15(t 20) a) What ticket price yields the maximum profit? b) What is the maximum profit? 5. Jason is standing on the ground and throws a ball vertically upward with an initial speed of 80 ft/sec. Its height after t seconds is given by h = 80t 16t 2 a) How high does the ball go? b) How many seconds does it take to reach maximum height? 13 c) When does the ball hit the ground?

14 6. Kevin in standing in a field and shoots an arrow vertically upward with an initial 14 speed of 64 ft/sec. Its height after t seconds is given by h = -16(x 2) a) How many seconds does it take to reach maximum height? b) How high does the arrow go? c) When does the arrow hit the ground? 7. Cory is standing on a cliff that is 48 feet above the ground.he throws a rock into the air. The height, h, of the rock after t seconds is given by h = 2(t 1) a) How many seconds does it take to reach maximum height? b) What is the rock s maximum height? c) How many seconds does it take for the rock to land on the ground? 8. Roger is standing at the top of a lighthouse. He is 96 feet above the ocean. He throws a ball into the air. The height, h, of the ball after t seconds is given by h = 3t t + 96 a) What is the ball s maximum height? b) How many seconds does it take to reach maximum height? c) How many seconds does it take for the ball to land in the ocean? 14

15 Systems of Quadratics 15 If you were to graph a quadratic function and a linear function, there are 3 possibilities. Example 1. solve this system algebraically. y = x 2 x 6 y = 2x 2 Try #1. Solve the system. y = 2x 2 +14x-15 y = 3x

16 You Try #2. y = x 2 +4x y = 2x + 6 Example #2. Solve the system. x 2 + y 2 = 26 x y = 6 You Try #3. Solve the system. x 2 + y 2 = 25 4y = 3x 16

17 Example #3. A rocket is launched from the ground and follows a parabolic path represented by the equation y = x x. At the same time, a flare is launched from a height of 10 feet and follows a straight path represented by the equation y = x Find the coordinates of the point or points where the paths intersect. 17 You Try #4. A pelican flying in the air over water drops a crab from a height of 30 feet. The distance the crab is from the water as it falls can be represented by the function h(t) = 16t , where t is time, in seconds. To catch the crab as it falls, a gull flies along a path represented by the function g(t) = 8t Can the gull catch the crab before the crab hits the water? 17

18 Practice and Closure Solve the following systems of equations 18 18

19 Systems of Quadratics 19 Solve the system 1) x 2 y = -2 2) x 2 + y = 1 -x + y = 4 2x + y = 2 3) x 2 + y = -5 4) x 2 + y 2 = 1 -x + y = 3 x 2 y = -1 5) x 2 + y 2 = 3 6) x 2 + y 2 = -6 4x 2 + y = 0-2x 2 + y = 7 19

20 Solve the system by graphing 20 11) The parabola y = x 2 6x ) The circle x 2 + y 2 = 5 and and the line y = -x + 7 the line y = x Practice and Closure Solve the systems 20

21 Quadratic Inequalities Review from Previous Math course: 21 Solve and graph the inequality 3x Solve and graph the inequality -2x 21 < 17 Quadratic Inequalities Example #1. Find the solution to the quadratic Inequality. x 2 + x 12-6 Step 1: Make sure it is in Step 2: Do the to find the Step 3: Step 4: Pick numbers for the 3 intervals and add them to the number line Step 5: Test all three numbers to see which answer(s) works First Interval Second Interval Third Interval If middle interval works : < x < If end intervals work: x < or x > 21

22 Example #2. Find and graph the solution. 22 x x + 25 > 9 You Try #1. Find and graph the solution. x 2 x + 24 > 36 Example #3. The annual profit, p(x), in dollars of a company varies with the number of employees, x, as p(x)= -40x x. What is the range of the number of employees for which the company s annual profit will be at least $112,000? 22

23 Example #4. The profit a coat manufacturer makes each day is modeled by the equation 2 P x 120x 2000, where P is the profit and x is the price for each coat sold. For what values of x does the company make a profit? 23 You Try #1. When a baseball is hit by a batter, the height of the ball, H, at time t, (t 0), is determined by the equation H 16t 2 64t 4. For which interval of time is the height of the ball greater than or equal to 52 feet? 2 You Try #2 The height of a punted football can be modeled by the function H 4.9x 20x 1, where H is given in meters and the time x is in seconds. At what time in its flight is the ball within 5 meters of the ground? 23

24 Practice and Closure The height of a ball above the ground after it is thrown upwards at 40 feet per second can be modeled by H 40x 16x 2 the function, where the height, H, is given in feet and the time x is in seconds. At what time in its flight is the ball within 15 feet of the ground? 2. A rectangle is 6 cm longer than it is wide. Find the possible dimensions if the area of the rectangle is more than 216 square centimeters. 3. Karen wants to plant a garden and surround it with decorative stones. She has enough stones to enclose a rectangular garden with a perimeter of 68 feet, but she wants the garden to cover no more than 240 square feet. What could the width of her garden be? 4. A manilla rope used for rappelling down a cliff can safely support a weight W (in pounds) provided 2 W 1480d Where d is the rope s diameter (in inches). What diameter of rope would be needed to support a weight of at least 5920 pounds? 24

25 Cartesian Island 25 Legend has it that long ago a Math Pirate named Euclid roamed the seas hiding his treasures. After many years and many riches, Pirate Euclid settled on a hidden island named Cartesian Island in the middle of Cartesian Ocean. He lived out his life with his first mate Descartes in peace and quiet spending his time contemplating geometry, calculus, and creating math puzzles. After his death, Descartes found his captains last math puzzle, a treasure map, identifying the burial site of his most precious and expensive treasure buried right on Cartesian Island. Use the clues below to help recreate the map and find Pirate Euclid s most prized and most precious treasure. Cartesian Island is a mostly parabolic shaped island with the top part enclosed by a radical function and a line. There are three main intersection points as well as the vertex of the parabola. These are key points of interest on the island. Graph the equations below to find the shape of the island: Quadratic: Q(x) =.125x 2.25x 10 Line: L(x) = 1 x Radical: R(x) = 2 x Vertex Pointe is located at the most southern tip of the island. Decimal Pointe is located in the west of the island where the quadratic meets the radical. Radical Ridge is located at the northern most part of the island where the radical meets the line. Original Pointe in the east where the line meets the quadratic. Cartesian Island is divided into three main sections by color. 25 The Pirate Paradise (purple) is located from x < 1. The Heel Haven (blue) is located from y > 7, The Pack Promise-land (red) is located between x >1 and y < 7.

26 There are four houses located on Cartesian s Island. 26 Mangum Mansion is located at the midpoint between Radical Ridge and Original Point. Mangum Mansion is located at (plot & label the point) Blackwell s Butterfly Bungalow is located at R(-1). Find the coordinates. Blackwell s Bungalow is located at (plot & label the point) Bello s Big Building is located at the right-most x-intercept of Q(x) Bello s Big Building is located at (plot & label the point) Driscoll s Dwelling is located at the left-most x-intercept of Q(x) Driscoll s Dwelling is located at (plot & label the point) There are only two roads on Cartesian Island. The main road, Spartan Street, from Decimal Point to Bello s Big Building. Equation of Spartan Street The second road, SohCahToa Street goes through the points (2,-5) and (5,4). Equation of SohCahToa Street The treasure is located at an alternate interior angle from radical ridge. It is located in the Pack Promise-Land. Label the location on the map. Find the intersection of those paths and you will find the treasure! If the x location is 5, what would the y value be? Location of Euclid s Treasure 26

27 Unit 7 Review 27 Graphing Quadratics Review Worksheet Fill in each blank using the word bank. vertex minimum axis of symmetry x-intercepts parabola maximum zeros/roots ax 2 + bx + c 1. Standard form of a quadratic function is y = 2. The shape of a quadratic equation is called a When the vertex is the highest point on the graph, we call that a. 6. When the vertex is the lowest point on the graph, we call that a. 7. Our solutions are the. 8. Solutions to quadratic equations are called. 27

28 Determine whether the quadratic functions have two real roots, one real root, or no real roots. If possible, list the zeros of the function Number of roots: 10. Number of roots: 11. Number of roots: Zero(s): Zero(s): Zero(s): 12. Given the graph, identify the following. Axis of symmetry: Vertex: How many zeros: which are: Domain: Range: 28

29 13. Graph the following quadratic functions by using critical values and/or factoring. You need three points to graph and don t necessarily need all the 29 information listed. y = x 2 2x - 3 Identify the zeros/roots: and Does it have a minimum or maximum? Axis of symmetry: Vertex: y-intercept: Domain: Range: Graph at least 5 points 14. y = -x 2 4x + 5 Does it have a minimum or maximum? Domain: y-intercept: Identify the zeros/roots and Axis of symmetry: Range: Vertex: 15. y = x 2 + 4x + 7 Does it have a minimum or maximum? Domain: y-intercept: Axis of symmetry: Range: Vertex: 16. y = -x 2-2x + 2 Does it have a minimum or maximum? Domain: y-intercept: Axis of symmetry: Range: Vertex: 29