5.2 Solving Linear-Quadratic Systems

Transcription

1 Name Class Date 5. Solving Linear-Quadratic Sstems Essential Question: How can ou solve a sstem composed of a linear equation in two variables and a quadratic equation in two variables? Resource Locker A..C: Solve, algebraicall, sstems of two equations in two variables consisting of a linear equation and a quadratic equation. Also A..A, A..D Eplore Investigating Intersections of Lines and Graphs of Quadratic Equations There are man real-world situations that can be modeled b linear or quadratic functions. What happens when the two situations overlap? Eamine graphs of linear functions and quadratic functions and determine the was the can intersect. Eamine the two graphs below to consider the was a line could intersect the parabola B Sketch three graphs of a line and a parabola: one in which the intersect in one point, one in which the intersect in two points, and one in which the do not intersect A constant linear function and a quadratic function can intersect at points. Module 5 55 Lesson

2 Reflect 1. If a line intersects a circle at one point, what is the relationship between the line and the radius of the circle at that point?. Discussion If a line is not horizontal, at how man points can it intersect a parabola? Eplain 1 Solving Linear-Quadratic Sstems Graphicall Graph each equation b hand and find the set of points where the two graphs intersect. Eample 1 Solve the given linear-quadratic sstem graphicall. - = + 6 = ( + 1) Solve each equation for. Plot the line and the parabola. - = = = ( + 1) = ( + 1) Find the approimate points of intersection: Estimating from the graph, the intersection points appear to be near (-1.5, -5.5) and (0.5, -.5). The eact solutions (which can be found algebraicall) are -1 - ( _, - _ - ) and ( -1 +, _ - ), or about (-1.7, -5.7) and (0.7, -.7). B + =.5 = ( - ) Solve each equation for. + =.5 = = Plot the line and the parabola on the aes provided Find the approimate point(s) of intersection:. Note that checking these coordinates in the original sstem shows that this is an eact solution. Module 5 56 Lesson

3 Your Turn Solve the given linear-quadratic sstem graphicall. + = = = - = 6 ( - ) Eplain Solving Linear-Quadratic Sstems Algebraicall Use algebra to find the solution. Use substitution or elimination. Eample Solve the given linear-quadratic sstem algebraicall. - = 7 + = ( + 5) Solve this sstem using elimination. First line up the terms. 7 + = + = ( + 5) Subtract the second equation from the first to eliminate the variable. Solve the resulting equation for using the quadratic formula. There is no real number equivalent to -15, so the sstem has no solution. 7 + = - ( + = ( + 5) = 0 = - ( + 5) = - ( + 5) = - ( ) = = = -17 ± 17-5 ) -17 ± 9 - = -17 ± -15 = Module 5 57 Lesson

4 B = _ 1 ( - ) - = 1 Solve the sstem b substitution. The first equation is alread solved for. Substitute the epression _ ( - ) for in the second equation. - ( ( - ) ) = 1 Now, solve for. 1 = - ( ( - ) ) 1 = - ( - ) 1 = - ( ) 1 = _ 1 = _ 0 = _ 5 0 = 0 = ( )( ) The line and the parabola intersect at two points. Use the -coordinates of the intersections to find the points. Solve - = 1 for. = ( )or = ( ) - = 1 - = 1 - = Find when = 5 and when = 7. = -_ 1-5 = -_ 1-7 = -_ 1-15 = -_ 1-1 = - _- = - _- = 1 = So the solutions to the sstem are. Reflect 5. How can ou check algebraic solutions for reasonableness? Module 5 5 Lesson

5 Your Turn Solve the given linear-quadratic sstem algebraicall. - 6 = - _ = = 7 - = 7 Eplain Solving Real-World Problems You can use the techniques from the previous eamples to solve real-world problems. Image Credits: jcsmill/ Shutterstock Eample Solve each problem. A tour boat travels around an island in an ellipical pattern. When the boat is directl north or south of the island, it is 6 units awa; and when it is directl east or west of the island, it is 5 units awa. Taking the island as (0, 0), a fishing boat approaches the island on a path that can be modeled b the equation - =. Is there a danger of collision? If so, where? Write the sstem of equations. The ellipse with vertical major ais of length 6 and horizontal minor ais of length 5 has the equation 5 + = 1. Multipling both sides 6 b 900, this is equivalent to = = = Solve the second equation for. - = = - = - _ Module 5 59 Lesson

6 Substitute for in the first equation = 900-6( _ ) + 5 = ( ) + 5 = ( - + 6) + 5 = = = 0 Solve using the quadratic equation. = 1 ± 1 - (1)(-6) (1) 1,000 = Collisions can occur when -.70 or To find the -values, substitute the -values into =. (-.70) - = = _ (5.) - = = 1 ± -.70 or 5. _ = _ _-1.0 = _ So the boats could collide at approimatel (-.7, -.70)or (1.1, 5.). Module 5 60 Lesson

7 B The signal from a radio station can be detected up to 5 units of distance awa. Taking the location of the station as (0, 0), a stretch of highwa near the station is modeled b the equation - 15 = 1. At which points, if an, does a car on the highwa enter and eit the 0 broadcast range of the station? Write the sstem of equations. With the station at (0, 0), the signal will reach out in a circle. The equation of a circle with radius 5 is + = 5 = = = 0 Solve the second equation for = 0 = Substitute for in the first equation. + ( ) + = 05 = 05 + = 05 _ _ + 5 = 05 _ - = = 0 Solve using the quadratic formula. = -600 ± (01)(-70000) (01) or (rounded to the nearest hundredth) To find the -values, substitute the -values into = The car will be within the radio station s broadcast area between. Module 5 61 Lesson

8 Your Turn. An asteroid is traveling toward Earth on a path that can be modeled b the equation = 1-7. It approaches a satellite in orbit on a path that can be modeled b the equation 9 + = 1. What are the 51 approimate coordinates of the points where the satellite and asteroid might collide? 9. The owners of a circus are planning a new act. The want to have a trapeze artist catch another acrobat in mid-air as the second performer comes into the main tent on a zip-line. If the path of the trapeze can be modeled b the parabola = _ + 16 and the path of the zip-line can be modeled b = + 1, at what point can the trapeze artist grab the second acrobat? Elaborate 10. A parabola opens to the left. Identif an infinite set of parallel lines that each of which intersect the parabola onl once. 11. If a parabola can intersect the set of lines = a a R in 0, 1, or points, what do ou know about the parabola? 1. Essential Question Check-In How can ou solve a sstem composed of a linear equation in two variables and a quadratic equation in two variables? Module 5 6 Lesson

9 Evaluate: Homework and Practice 1. How man points of intersection are on the graph?. How man points of intersection are there on the graph of = + -? - = Solve each given linear-quadratic sstem graphicall. If necessar, round to the nearest integer.. = -( - ) +. - = ( - 1) = -5 + = = = ( + 1) - + = = Module 5 6 Lesson

10 7. ( - ) + = = = _ Solve each linear-quadratic sstem algebraicall = = = ( - ) + = = - - = = - + = 1 1. = = 1. = = Module 5 6 Lesson

11 Write and solve a sstem of equations to find the solutions. 15. Jason is driving his car on a highwa at a constant rate of 60 miles per hour when he passes his friend Alan whose car is parked on the side of the road. Alan has been waiting for Jason to pass so that he can follow him to a nearb campground. To catch up to Jason's passing car, Alan accelerates at a constant rate. The distance d, in miles, that Alan's car travels as a function of time t, in hours, since Jason's car has passed is given b d = 600t. How long does it takes Alan's car to catch up with Jason's car? 16. The flight of a cannonball toward a hill is described b the parabola = The hill slopes upward, at a grade of 15% (meaning it rises 15 feet for ever 100 horizontal feet it runs). Where on the hill does the cannonball land? Image Credits: Corbis 17. A ball is launched into the air from the ground with an initial vertical velocit of 6 ft/sec. At the same time a balloon is released from a height of 0 feet, and it rises at the rate of ft/sec. Write and solve a sstem of equations to determine when the ball and the balloon are at the same height. Module 5 65 Lesson

12 1. The range of an ambulance service is twent miles in an direction from its station. Taking the ambulance's station to be at (0, 0), a straight road within the service area is represented b = + 0. Find the length in miles of the road that lies within the range of the ambulance service (round our answer to the nearest hundredth). Recall that the distance formula is d = ( - 1 ) + ( - 1 ). 19. Match the equations with their solutions. = = - A. (, ) (-, -) = ( - ) B. (0, -) (5, ) = -5 - = C. (, 0) + = 5 = ( - ) D. No solution = 0 0. A student solved the sstem - 7 = - 5 graphicall and - = 1 determined the onl solution to be (1, ). Was this a reasonable answer? How do ou know? Image Credits: Glen Jones/Shutterstock - Module 5 66 Lesson

13 H.O.T. Focus on Higher Order Thinking 1. Eplain the Error A student was asked to come up with a sstem of equations, one linear and one quadratic, that has two solutions. The student gave = -( + 1) + 9 = - + as the answer. What did the student do wrong?. Analze Relationships The graph shows a quadratic function and a linear function = d. If the linear function were changed to = d +, how man solutions would the new sstem have? If the linear function were changed to = d - 5, how man solutions would the new sstem have? Give reasons for our answers Make a Conjecture Given = 100 and = , what can ou sa about an line that goes through the verte of each but is not horizontal or vertical?. Communicate Mathematical Ideas Eplain wh a sstem of a linear equation and a quadratic equation cannot have an infinite number of solutions. Module 5 67 Lesson

14 Lesson Performance Task Suppose an aerial freestle skier goes off a ramp with her path represented b the equation = -0.0( - 5 ) + 0. If the surface of the mountain is represented b the linear equation = , find the distance in feet the skier lands from the beginning of the ramp. Image Credits: EpicStockMedia/Alam Module 5 6 Lesson