Name Class Date 16-4 Nonlinear Sstems Going Deeper Essential question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? To estimate the solution to a sstem of equations, ou can graph both equations on the same coordinate plane and find the intersection points. Or ou can solve the equations algebraicall using substitution or elimination. Video Tutor 1 MCC9 1.A.REI.7 EXAMPLE Solving b Graphing and Algebraicall Solve the sstem of equations. f() = -8 + 48 g() = -( - ) + 3 A Solve the sstem of equations b graphing. Start b graphing the quadratic function. The verte is (, ). Describe the transformation of the parent quadratic function that produces the graph of g(). To make the graph more accurate, plot the points where the -intercepts occur. The -intercepts are the solutions of the equation g() = 0: - ( - ) + 3 = 0 - ( - ) = ( - ) = - = ± = ± = or So, the points (, 0 ) and (, 0 ) are on the graph. Use these points and the verte to draw the graph. Now graph the linear function. The -intercept is, and the slope is. 40 3 4 16 8-4 - 0 4 6 The line and the parabola intersect at two points. Identif the coordinates of those points. (, ) and (, ) Module 16 375 Lesson 4
B Solve the sstem of equations algebraicall. Write the functions in terms of. = -8 + 48 = -( - ) + 3 Both equations are solved for, so set the right sides equal to each other and solve for. -8 + 48 = -( - ) + 3-8 + 48 = Simplif the right side. 8-48 = 8-48 Add 8-48 to both sides. 0 = Simplif both sides. 0 = - ( ) ( ) Factor the right side. = or = Use the zero-product propert to solve for. Substitute these values of into the equation of the line to find the corresponding -values. = -8() + 48 = = -8(6) + 48 = The solutions are (, ) and (, ). REFLECT 1a. If the linear function was f () = 8 + 48, how man solutions would there be? Justif our answer. 1b. When solving algebraicall, wh do ou substitute the -values into the equation of the line instead of the equation of the parabola? 1c. Eplain the relationship between the intersection points of the graphs and the solutions of the sstem of equations. 1d. Describe how to check that the solutions are correct. Module 16 376 Lesson 4
MCC9 1.A.REI.7 eplore Determining the Possible Number of Solutions In the previous eample, the sstem of equations had two solutions. You can use a graph to understand other possible numbers of solutions of a sstem of equations involving a linear equation and a quadratic equation. The graph of the quadratic function f() = - + 10-7 is shown below. Graph each linear function below on the same coordinate plane as the parabola. Line 1: g() = - 11 Line : h() = - + 14 1 8 4-4 0 4 6 8 10-8 REFLECT a. At how man points do the parabola and Line 1 intersect? How man solutions are there for the sstem consisting of the quadratic function and the first linear function? b. At how man points do the parabola and Line intersect? How man solutions are there for the sstem consisting of the quadratic function and the second linear function? c. A sstem of equations consisting of one quadratic equation and one linear equation can have,, or real solutions. d. How man solutions does the following sstem of equations have? Eplain our reasoning. f () = - + 10-7 k() = - + 1 e. How man solutions does the following sstem of equations have? Eplain our reasoning. f () = - + 10-7 p() = - Module 16 377 Lesson 4
You can use the Intersect feature on a graphing calculator to solve sstems of equations. 3 MCC9 1.A.REI.7 EXAMPLE Solving Sstems Using Technolog Use a graphing calculator to solve the sstem of equations. f() = -4.9 + 50 + 5 g() = 30 A Enter the functions as Y 1 and Y on a graphing calculator. Then graph both functions. Sketch the graphs on the coordinate plane at the right. Estimate the solutions of the sstem from the graph. 160 10 80 40 B Solve the sstem directl b using the Intersect feature of the graphing calculator. -4 0 4 8 1-40 Press nd and CALC, then select Intersect. Press Enter for the first curve and again for the second curve. For Guess?, press the left or right arrows to move the cursor close to one of the intersections, then press Enter again. Repeat, moving the cursor close to the other intersection to find the second solution. Round our solutions to the nearest tenth. REFLECT 3a. Are the solutions ou get using the Intersect feature of a graphing calculator alwas eact? Eplain. 3b. How can ou check the accurac of our estimated solutions? 3c. Use a graphing calculator to solve the sstem of equations f () and h() where h() = 30 + 50. What is the result? Eplain. Module 16 378 Lesson 4
practice Solve the sstem of equations algebraicall. Round to the nearest tenth, if necessar. 1. f () = - g() = -. = ( - 3 ) = 3. = - - 4 + 1 = - 1 + 3 4. f () = g() = 1 5. = + 4-5 = 3-6. f () = -16 + 15 + 10 g() = 14 - The graph of a sstem of equations is shown. State how man solutions the sstem has. Then estimate the solution(s). 7. 8. 0 5 10-8 -6-4 - 0 4-10 -0-30 -40 4 3 1-1 0 1 3 4 5-1 Module 16 379 Lesson 4
Estimate the solutions to the sstem of equations graphicall. Confirm the solutions b substituting the values into the equations. 9. f () = g() = 1 10. = - 1 = 0.5-3 11. f () = -16 + 15 + 10 g() = 14-1. f () = 3( - 1 ) + 4 g() = -4 + 9 Solve the sstem of equations using the Intersect feature of a graphing calculator. Round our answers to the nearest tenth. 13. = - + 6 + 7 = + 6 14. f () = - + - g() = - 3 Module 16 380 Lesson 4
Name Class Date Additional Practice 16-4 Solve each sstem b graphing. Check our answers. 1.. 3 6 _ Solve each sstem b substitution. Check our answers. 4 3 3. 4. 5 8 6 _ 5. 5 3 1 6. 16 4 _ 7. 9. 1 8 _ 3 1 4 _ 8. 10. 3 4 4 3 4 Module 16 381 Lesson 4
Problem Solving Write the correct answer. 1. A ball is thrown upward with an initial velocit of 40 feet per second from ground level. The height h in feet of the ball after t seconds is given b h 16t 40 t. At the same time, a balloon is rising at a constant rate of 10 feet per second. Its height h in feet after t seconds is given b h 10t. Find the time it takes for the ball and the balloon to reach the same height. 3. A skateboard compan's monthl sales income can be modeled b the equation Cs ( ) 0.5s 5s 500, where s represents the number of skateboards sold. The monthl cost of running the business is Cs ( ) 5s 81.5. How man skateboards must the compan sell in a month before the sales income equals or eceeds the cost of running the business? Select the best answer. 5. A seagull is fling upwards such that its height h in feet above the sea after t seconds is given b h 3t. At the same time, the height h in feet of a rock falling off a cliff above the sea after t seconds is given b h 16t 50. Find the approimate time it takes for the rock and the bird to be at the same height. A 1.68 seconds C 3.36 seconds B 3.13 seconds D 16.67 seconds. A bird starts fling up from the grass in a park and climbs at a stead rate of 0.5 meters per second. Its height h in meters after t seconds is given b h 0.5t. The equation h 4.9t 40t 3 models the height h, in meters, of a baseball t seconds after it is hit. Find the time it takes for the ball and the bird to reach the same height. 4. The deer population in a park can be modeled b the equation P ( ) 4 10 60, where is the number of ears after 010. The deer population in another park can be modeled b P ( ) 10 80, where is the number of ears after 010. In which ear will the two parks have approimatel the same number of deer? 6. A juggler at a fun park throws a ball upwards such that the ball's height h in feet above the ground after t seconds is given b h 16t 0t 5. At the same time, a scenic elevator begins climbing a tower at a constant rate of 0 feet per second. Its height h in feet after t seconds is given b h 0t. Find the approimate time it takes for the ball and the elevator to reach the same height. F 0.56 seconds H 4 seconds G 1.1 seconds J 11.18 seconds Module 16 38 Lesson 4