9.5 Solving Nonlinear Systems

Transcription

1 Name Class Date 9.5 Solving Nonlinear Sstems Essential Question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? Eplore Determining the Possible Number of Solutions of a Sstem of Linear and Quadratic Equations A sstem of one linear and one quadratic equation ma have zero, one, or two solutions. Resource Locker A The graph of the quadratic function ƒ() = - - is shown. On the same coordinate plane, graph the following linear functions: g ()= - -, h ()= - 6, j () = Houghton Mifflin Harcourt Publishing Compan B C D Look at the graph of the sstem consisting of the quadratic function, ƒ (), and the linear function, g (). Based on the intersections of these two graphs, how man solutions eist in a sstem consisting of these two functions? Look at the graph of the sstem consisting of the quadratic function, ƒ (), and the linear function, h (). Based on the intersections of these two graphs, how man solutions eist in a sstem consisting of these two functions? Look at the graph of the sstem consisting of the quadratic function, ƒ (), and the linear function, j (). Based on the intersections of these two graphs, how man solutions eist in a sstem consisting of these two functions? Reflect 1. A sstem consisting of a quadratic equation and a linear equation can have,, or solutions. Module 9 11 Lesson 5

2 Eplain 1 Solving a Sstem of Linear and Quadratic Equations Graphicall A sstem of equations consisting of a linear and quadratic equation can be solved graphicall b finding the points where the graphs intersect. Eample 1 Solve the sstem of equations graphicall. = ( + 1) - A = - Graph the quadratic function. The verte is the point ( 1, ). The -intercepts are the points where = 0. ( + 1) - = 0 ( + 1) = + 1 = ± = 1 or = -3 Graph the linear function on the same coordinate plane. The solutions of the sstem are the points where the graphs intersect. The solutions are ( 1, ) and (1, 0). = ( - ) B - = Graph the quadratic function. The verte is the point,. The -intercepts are the points where = 0. ( - ) - = 0 ( - ) = ( - ) = = ±1 = or = Graph the linear function on the same coordinate plane. There are intersection points. This sstem has solution(s) Houghton Mifflin Harcourt Publishing Compan Module 9 1 Lesson 5

3 Your Turn Solve the sstem of equations graphicall.. = -( + ) + 8 = = ( + 1) - 9 = Eplain Solving a Sstem of Linear and Quadratic Equations Algebraicall Sstems of equations can also be solved algebraicall b using the substitution method to eliminate a variable. If the sstem is one linear and one quadratic equation, the equation resulting after substitution will also be quadratic and can be solved b selecting an appropriate method. Eample Solve the sstem of equations algebraicall. Houghton Mifflin Harcourt Publishing Compan = ( + 1) - A = - Set the two the epressions for equal to each other, and solve for. ( + 1) - = = = 0 = 1 = ±1 Substitute 1 and -1 for to find the corresponding -values. = - = - = (1) - = 0 = (-1) - = - The solutions are (1, 0) and (-1, -). Module 9 13 Lesson 5

4 B = ( + )( + 1) = Set the two the epressions for equal to each other, and solve for. ( + )( + 1) = + + = = 0 ( + ) ( + 3) = 0 = Substitute -3 for to find the corresponding -value. = = -( ) - 5 = The solution is. Reflect. Discussion After finding the -values of the intersection points, wh use the linear equation to find the -values rather than the quadratic? What if the quadratic equation is used instead? Your Turn Solve the sstem of equations algebraicall. 5. = = 3-3 Houghton Mifflin Harcourt Publishing Compan Module 9 1 Lesson 5

5 Eplain 3 Solving a Real-World Problem with a Sstem of Linear and Quadratic Equations Sstems of equations can be solved b graphing both equations on a graphing calculator and using the Intersect feature. Eample 3 Create and solve a sstem of equations to solve the problem. A A rock climber is pulling his pack up the side of a cliff that is feet tall at a rate of feet per second. The height of the pack in feet after t seconds is given b h = t. The climber drops a coil of rope from directl above the pack. The height of the coil in feet after t seconds is given b h = -16 t At what time does the coil of rope hit the pack? Create the sstem of equations to solve. h = -16t h = t Graph the functions together and find an points of intersection. Houghton Mifflin Harcourt Publishing Compan Image Credits: Andresr/ Shutterstock B The intersection is at (-3.375, -6.75). The intersection is at (3.5, 6.5). The -value represents time, so this solution is not reasonable. This solution indicates that the coil hits the pack after 3.5 seconds. A window washer is ascending the side of a building that is 50 feet tall at a rate of 3 feet per second. The elevation of the window washer after t seconds is given b h = 3t. The supplies are lowered to the window washer from the top of the building at the same time that he begins to ascend the building. The height of the supplies in feet after t seconds is given b h = - t At what time do the supplies reach the window washer? Create the sstem of equations to solve. h = t + h = t Module 9 15 Lesson 5

6 Graph the functions together and find an points of intersection. The intersection is at about,. The intersection is at about The -value represents, so this This solution indicates that solution is.,. Reflect 6. How did ou know which intersection to use in the eample problems? Your Turn Write and solve a sstem of equations to solve the problem. 7. A billboard painter is using a pulle sstem to hoist a can of paint up to a scaffold at a rate of half a meter per second. The height of the can of paint as a function of time is given b h (t) = 0.5t. Five seconds after he starts raising the can of paint, his partner accidentall kicks a paint brush off of the scaffolding, which falls to the ground. The height of the falling paint brush can be represented b h (t) = -.9 (t - 5) When does the brush pass the paint can? Houghton Mifflin Harcourt Publishing Compan Module 9 16 Lesson 5

7 Elaborate 8. Discussion When solving a sstem of equations consisting of a quadratic equation and a linear equation b graphing, wh is it difficult to be sure there is one solution as opposed to 0 or? 9. How can ou use the discriminant to determine how man solutions a linear-quadratic sstem has? 10. Essential Question Check-in How can the graphs of two functions be used to solve a sstem of a quadratic and a linear equation? Evaluate: Homework and Practice Houghton Mifflin Harcourt Publishing Compan 1. The graph of the function ƒ () = - 1_ ( - 3) + is shown. Graph the functions g () = + 1, h () = +, and j () = + 3 with the graph of ƒ (), and determine how man solutions each sstem has. ƒ () and g () : ƒ () and h () : ƒ () and j () : Solve each sstem of equations graphicall.. = ( + 3) - = = - 1 = Online Homework Hints and Help Etra Practice - - Module 9 17 Lesson 5

8 . = ( - ) - 5. = - + = - = = -( - ) = 3( + 1) - 1 = = Solve the sstem of equations algebraicall. 8. = = = 5 = = ( - 3) = 11. = - + = + Houghton Mifflin Harcourt Publishing Compan Module 9 18 Lesson 5

9 1. = = = + 7 = = = = ( + )( + ) = 3 + Create and solve a linear quadratic sstem to solve the problem. Houghton Mifflin Harcourt Publishing Compan Image Credits: Germanskdiver/Shutterstock 16. The height in feet of a skdiver t seconds after deploing her parachute is given b h(t)= -300t A ball is thrown up toward the skdiver, and after t seconds, the height of the ball in feet is given b h(t)= -16t + 100t. When does the ball reach the skdiver? 17. A wildebeest fails to notice a lion that is charging from behind at 65 feet per second until the lion is 0 feet awa. The lion s position as a function of time is given b p(t) = 65t - 0. The wildebeest has to begin accelerating from a standstill until it is captured or reaches a top speed fast enough to sta ahead of the lion. The wildebeest s position as a function of time is given b d(t) = 35t. Does the wildebeest escape? Module 9 19 Lesson 5

10 18. An elevator in a hotel moves at 0 feet per second. Leaving from the ground floor, its height in feet after t seconds is given b the formula h(t) = 0t. A bolt comes loose in the elevator shaft above, and its height in feet after falling for t seconds is given b h(t) = -16t At what time and at what height does the bolt hit the elevator? 19. A bungee jumper leaps from a bridge 100 meters over a gorge. Before the 0-meter-long bungee begins to slow him down, his height is characterized b h(t) = -.9t Two seconds after he jumps, a car on the bridge blows out a tire. The sound of the tire blow-out moves down from the top of the bridge at the speed of sound and has a height given b h(t)= -30(t - ) How high will the bungee jumper be when he hears the sound of the blowout? 0. Eplain the Error A student is asked to solve the sstem of equations = and - = + 1. For the first step, the student sets the right hand sides equal to each other to get the equation = + 1. Wh does this not give the correct solution? 1. Eplain the Error After solving the sstem of equations in Eercise 18 (the elevator and the bolt), a student concludes that there are two different times that the bolt hits the elevator. What is the error in the student s reasoning?. Multi-part Classification The functions listed are graphed here. f 1 ()= ( + 3) + 1 and f ()= - 3_ ( - ) + 3 g 1 ()= + 3 and g ()= 3 and g 3 ()= - 1_ + 1 Use the graph to classif each sstem as having 0, 1, or solutions. a. = ƒ 1 () b. = g 1 () = ƒ 1 () c. = g () = ƒ 1 () = g 3 () - 0 Houghton Mifflin Harcourt Publishing Compan d. = ƒ () e. = g 1 () = ƒ () f. = g () = ƒ () = g 3 () Module 9 0 Lesson 5

11 H.O.T. Focus on Higher Order Thinking 3. Eplain the Error After solving the sstem of equations in Eercise 16 (the skdiver and the ball), a student concludes there are two valid solutions because the both have positive times. The ball must pass b the skdiver twice. What is the error in the student s reasoning?. Multi-Part Problem The path of a baseball hit for a home run can be modeled b = - _ + + 3, where and are in feet and home plate is at the origin. 8 The ball lands in the stands, which are modeled b - = -35 for 00. Use a graphing calculator to graph the sstem. a. What do the variables and represent? b. About how far is the baseball from home plate when it lands? c. About how high up in the stands does the baseball land? 5. Draw Conclusions A certain sstem of a linear and a quadratic equation has two solutions, (, 7)and (5, 10). The quadratic equation is = What is the linear equation? Justif our answer. Houghton Mifflin Harcourt Publishing Compan 6. Justif Reasoning It is possible for a sstem of two linear equations to have infinitel man solutions. Eplain wh this is not possible for a sstem with one linear and one quadratic equation. Module 9 1 Lesson 5

12 Lesson Performance Task A race car leaves pit row at a speed of 0 feet per second and accelerates at a constant rate of feet per second squared. Its distance from the pit eit is given b the function d r (t)= t + 0t. The race car leaves ahead of an approaching pace car traveling at a constant speed of 10 feet per second. In each case, find out if the pace car will catch up to the race car, and if so, how far down the track it will catch up. If there is more than one solution, eplain how ou know which one to select. a. The pace car passes b the eit to pit row 1 second after the race car eits. b. The pace car passes the eit half a second after the race car eits. Houghton Mifflin Harcourt Publishing Compan Module 9 Lesson 5