8-10 Nonlinear Sstems CC.9-1.A.REI.7 Solve a simple sstem consisting of a linear equation and a quadratic equation in two variables algebraicall and graphicall. Objective Solve sstems of equations in two variables in which one equation is linear and the other is quadratic. Vocabular nonlinear sstem of equations Wh learn this? You can solve a nonlinear sstem to find how long it takes for two objects to reach the same height. (See Eample.) Recall that a sstem of linear equations is a set of two or more linear equations. A solution of a sstem is an ordered pair that satisfies each equation in the sstem. Points where the graphs of the equations intersect represent solutions of the sstem. A nonlinear sstem of equations is a sstem in which at least one of the equations is nonlinear. For eample, a sstem that contains one quadratic equation and one linear equation is a nonlinear sstem. A sstem made up of a linear equation and a quadratic equation can have no solution, one solution, or two solutions, as shown below. No solution One solution Two solutions EXAMPLE 1 Solving a Nonlinear Sstem b Graphing A quadratic function has the form = a + b + c. To graph a quadratic function, start b using = - b a to find the ais of smmetr and the verte. Solve the sstem b graphing. Check our answer. = - - 3 = - - 1 Step 1 Graph = - - 3. The ais of smmetr is = 1. The verte is (1, -). The -intercept is -3. Another point is (-1, 0). Graph the points and reflect them across the ais of smmetr. Step Graph = - - 1. The slope is -1. The -intercept is -1. (-1, 0) Step 3 Find the points where the two graphs intersect. The solutions appear to be (-1, 0) and (, -3). = - - 3 - - = - - 1 (, -3) 590 Chapter 8 Quadratic Functions and Equations
Check Substitute (-1, 0) into the sstem. Substitute (, -3) into the sstem. = - - 3 = - - 1 = - - 3 = - - 1 0 (-1) - (-1)- 3 0 -(-1) - 1-3 - () - 3-3 - - 1 0 1 + - 3 0 1-1 -3 - - 3-3 -3 0 0 0 0-3 -3 The solutions are (-1, 0) and (, -3). 1. Solve the sstem b graphing. Check our answer. = - + 5 = + 1 EXAMPLE Solving a Nonlinear Sstem b Substitution The substitution method is a good choice when either equation is solved for a variable, both equations are solved for the same variable, or a variable in either equation has a coefficient of 1 or -1. Solve the sstem b substitution. = - 3 + = + = - 3 + Both equations are solved for. = + = - 3 + + = - 3 + -( + ) -( + ) = + = 1 + = 3 0 = - + 0 = ( - + 1) 0 = ( - 1)( - 1) The solution is (1, 3). 0; - 1 = 0 Check Use a graphing calculator. = 1 The graph supports the result found above. This sstem has eactl one real solution. The graph of the sstem consists of a line and a parabola that meet in eactl one point. Substitute + for in the first equation. Subtract + from both sides. Factor out the GCF,. Factor the trinomial. Use the Zero Product Propert; cannot equal 0. Solve the remaining equation. Write one of the original equations. Substitute 1 for.. Solve the sstem b substitution. Check our answer. = 3-3 + 1 = -3 + 8-10 Nonlinear Sstems 591
EXAMPLE 3 Solving a Nonlinear Sstem b Elimination The elimination method is a good choice when both equations have the same variable term with the same or opposite coefficients or when a variable term in one equation is a multiple of the corresponding variable term in the other equation. Solve each sstem b elimination. A B - = = + 1 - = + = + 1 = + 3 - - 0 = - + 3 0 = ( - 1)( - 3) - 1 = 0 or - 3 = 0 = 1 or = 3 = + 1 = + 1 = 1 + 1 = 3 + 1 = = 10 The solutions are (1, ) and (3, 10). = + - 1-3 = 6-3 = 6 = + - 1 3() = 3( + - 1) - 3 = 6 + - 3 = 6 3 = 3 + 3-3 = 3 + 3 + 3 0 = 3 + + 3 = -1 ± (1) - (3)(3) (3) -1 ± = 1-36 6-1 ± = -35 6 There are no real solutions. Check Use a graphing calculator. To graph - 3 = 6, first solve for. - 3 = 6-3 = - + 6 = _ 3 - The graph supports that there are no real solutions. Write the sstem to align the -terms. Add the equations to eliminate. Subtract from both sides. Factor the trinomial. Use the Zero Product Propert. Solve the equations. Write one of the original equations. Substitute each -value and solve for. Write the sstem to align the -terms. Multipl each term in the first equation b 3. Add the second equation to the new first equation to eliminate. Subtract from both sides. Use the Quadratic Formula, = -b ± b - ac. a Note the discriminant: b - ac = -35. Its value is negative, so there are no real solutions. 59 Chapter 8 Quadratic Functions and Equations
Solve each sstem b elimination. Check our answers. - = 3a. = - - 5 3b. = - 5 5 - = 5 EXAMPLE Phsics Application When t = 0, the ball and elevator are at the same height because the are both at ground level. An elevator is rising at a constant rate of 0 feet per second. Its height in feet after t seconds is given b h = 0t. At the instant the elevator is at ground level, a ball is thrown upward with an initial velocit of 80 feet per second from ground level. The height in feet of the ball after t seconds is given b h = -16t + 80t. Find the time it takes for the ball and the elevator to reach the same height. h = -16t Solve the sstem + 80t b substitution. h = 0t -16t + 80t = 0t - 0t -0t -16t + 60t = 0 -t(t - 15) = 0 -t = 0 or t - 15 = 0 t = 0 t = 15 Solve the remaining equations. t = 3.75 Substitute -16t + 80t for h in the second equation. Subtract 0t from both sides. Factor out the GCF, -t. Use the Zero Product Propert. It takes 3.75 seconds for the ball and the elevator to reach the same height.. An elevator is rising at a constant rate of 8 feet per second. Its height in feet after t seconds is given b h = 8t. At the instant the elevator is at ground level, a ball is dropped from a height of 10 feet. The height in feet of the ball after t seconds is given b h =-16t + 10. Find the time it takes for the ball and the elevator to reach the same height. THINK AND DISCUSS 1. How is solving the sstems in this lesson similar to solving sstems of linear equations? How is it different?. When using elimination to solve a linear/quadratic sstem, which variable will be eliminated? Wh? 3. A sstem of linear equations can have infinitel man solutions. Wh can t a linear/quadratic sstem have infinitel man solutions?. GET ORGANIZED Cop and complete the graphic organizer b sketching diagrams to show eamples. Write not possible for an cases that are not possible. Sstem of Equations Linear Linear/Quadratic Number of Solutions 0 1 infinite 8-10 Nonlinear Sstems 593
8-10 Eercises GUIDED PRACTICE Vocabular Appl the vocabular from this lesson to answer each question. 1. A sstem of equations that includes a linear equation and a quadratic equation is?. (linear, nonlinear, or quadratic). Sketch a nonlinear sstem of equations that has two solutions. The sstem should include one quadratic equation and one linear equation. SEE EXAMPLE 1 Solve each sstem b graphing. Check our answers. = - 7 + 6 3. = = - - 5. = - 8 SEE EXAMPLE Solve each sstem b substitution. Check our answers. = - + 3 5. = - 3 SEE EXAMPLE 3 Solve each sstem b elimination. Check our answers. = - 3 7. - = 6 = - 5 + 3 6. = -3 + 15 = + 7 + 1 8. 3 - = 5 SEE EXAMPLE 9. Phsics A bird is fling upwards such that its height in feet after t seconds is given b h = t. At the instant the bird passes the height of a ball being held out of a window, the ball is thrown upward with an initial velocit of 80 feet per second. The height in feet of the ball after t seconds is given b h = -16t + 80t. Find the time it takes for the ball and the bird to reach the same height. Independent Practice For See Eercises Eample 10 13 1 1 17 18 1 3 5 Etra Practice See Etra Practice for more Skills Practice and Applications Practice eercises. PRACTICE AND PROBLEM SOLVING Solve each sstem b graphing. Check our answers. = - 10. = - 3 11. = 5-10 - 6 = 18 = + + 7 1. = - 8 + 3 13. = + 5 = 6-1 Solve each sstem b substitution. Check our answers. = + 7 + 1. = - 3 15. = 5 + 5 = + 9 = - 5 16. = 5-17. = - + 3 = 10 + 9 Solve each sstem b elimination. Check our answers. = - 3 + 1 18. = - 5 19. 5 - = -1-3 = 15 = - + 7 0. + 3 = 6 = + 5 1. 9 - = 3 59 Chapter 8 Quadratic Functions and Equations
. Demographics The growing population of town A can be modeled b the equation P(t) = 8t + 000, where t represents number of ears after 010. The growing population of town B can be modeled b the equation P(t) = 100t + 3000. In which ear will the populations of the towns be approimatel equal? 3. Finance The value of Danielle s investments is modeled b the equation V(t) = 3t + 70t + 100, where t represents the number of months after she made her initial investment. Jeffre has no mone invested in stocks, but he deposits the same amount ever month into a savings account that he opened at the same time as Danielle began investing. His savings account balance can be modeled b the equation V(t) = 50t + 75. After how man months will the value of Danielle s investments be equal to the balance of Jeffre s savings account?. Amusement Parks A ride at an amusement park consists of an observation deck that travels directl up into the air at a constant rate of 0 feet per second. Its height in feet after t seconds is given b h = 0t. At the instant the deck is at ground level, a ball is thrown up with initial velocit 60 feet per second from ground level. The height in feet of the ball after t seconds is given b h = -16t + 60t. Find the time it takes for the ball and the deck to reach the same height. Round our answer to the nearest hundredth. 5. Business A compan s weekl revenue can be modeled b the equation C(p) = 0.75p + 10p + 00, where p represents the number of products sold. The weekl cost of running the business is modeled b the equation C(p) = 80p + 700. How man products must the compan sell in a week to break even (when revenue equal the costs of running the business)? Determine whether the point is a solution of the sstem of equations. = - 9 + 6. + = -16 ; (-, -1) 7. = + - 9 ; (3, ) = 8 = - 6-1 8. 3 - = -3 ; (7, 6) 9. = 3-7 + 6 ; (1, ) = + 1 = - 5 + 5 30. = 3 + - 1 ; (3, 8) 31. ; (-1, -) = + 5 9 - = -5 = - 5 3. = + 3 + 8 ; (5, 5) 33. ; (, -18) - = 5 5 - = 8 3. Write About It Eplain in our own words when ou should use the substitution method to solve a nonlinear sstem, and when ou should use the elimination method. 35. Critical Thinking Describe a scenario in which ou might use the graphing method to solve a sstem of nonlinear equations, even if ou didn t epect the solution(s) to consist of integer coordinates. Wolfgang Deuter/Corbis 36. Estimation Estimate the solution(s) to the sstem b graphing. = + 6 - = 0.5 + 7 8-10 Nonlinear Sstems 595
A 37. /ERROR ANALYSIS / Below are two solutions to the sstem of equations = - + 5. Which is incorrect? Eplain the error. + 3 = 11 = - + 5-3 = -3 + 1-15 + 3 = 11 + 3 = 11 + 0 = -3 + 1-0 = -3 + 8-0 = (-3 + )( - ) -3 + = 0 or - = 0-3 = - = = 3 + 3 = 11 + 3 = 11 ( ) 3 8 3 + 3 = 11 () + 3 = 11 + 3 = 11 8 + 3 = 11 3 5 = 3 3 = 3 5 = 9 = 1, ( 5 3 9 ) The solutions are and (, 1). B = - + 5-3 = - + 5 + 3 = 11 + 3 = 11 + 0 = - + 5 0 = - 8 + 5 = - b ± b - ac a = - (-8) ± (-8) - (1)(5) (1) = 8 ± = 8 ± 11 = ± 11 7.3 or 0.68 + 3 11 + 3 11 (7.3) + 3 11 (0.68) + 3 11 9.8 + 3 11.7 + 3 11 3-18.8 3 8.8-6.09.76 The solutions are approimatel (7.3, -6.09) and (0.68,.76). 38. Which are the solutions to the sstem of equations below? = - 5 - = -7 + 1 (-3, ) and (1, -6) (-1, 8) and (, -10) (-3, ) and (-1, 8) (1,-6) and (, -10) 39. For which sstem of equations is (, 6) a solution? = + - = 3 - - = -3 - = -5 + 16 = - + 1 = + 9 - = = 5-0. Which sstem below has no real solutions? 3 + = 9 = ( - 3) = - - = 5 = - = - - = - 7 = - 7 596 Chapter 8 Quadratic Functions and Equations
= - 1 1. Which is the graph of = + - 5? - 0 - - - - 0 - - - CHALLENGE AND EXTEND Find the -coordinate(s) of the solution(s) of each sstem. = 3 + - 7. = - 3-5 3. 3 + 5 = 8 3 + = 8 = + -. 3 + = 3 = - 9 5. = 7-6. A sstem of two equations contains one quadratic equation and one linear equation. The quadratic equation in the sstem is = + 5-9. The solutions of the sstem are (3, 15) and (-1, -13). What is the linear equation in the sstem? 7. Phsics The formula for the height of an object in free fall (neglecting air resistance) is h(t) = -16t + v 0 t + h 0, where v 0 is the object s initial velocit in feet per second and h 0 is the object s initial height above the ground in feet. One ball is thrown with an initial velocit of 90 ft/s from a height of 0 ft. A second ball is thrown at the eact same instant with an initial velocit of 80 ft/s and a height of 30 ft. After how man seconds will the balls reach the same height? 8-10 Nonlinear Sstems 597
CHAPTER SECTION 8B Make sense of problems and persevere in solving them. Solving Quadratic Equations Seeing Green A golf plaer hits a golf ball from a tee with an initial vertical velocit of 80 feet per second. The height of the golf ball t seconds after it is hit is given b h = - 16t + 80t. 1. How long is the golf ball in the air?. What is the maimum height of the golf ball? 3. How long after the golf ball is hit does it reach its maimum height?. What is the height of the golf ball after 3.5 seconds? 5. At what times is the golf ball 6 feet in the air? Eplain. (tl),photodisc/gett Images; (tc), COMSTOCK, Inc.; (tr),stuart Franklin/Gett Images; (b),robert Laberge/Gett Images 598 Chapter 8 Quadratic Function and Equations
CHAPTER Quiz for Lessons 8-5 Through 8-10 SECTION 8B 8-5 Solving Quadratic Equations b Graphing Solve each equation b graphing the related function. 1. - 9 = 0. + 3 - = 0 3. + 8 = 3. The height of a fireworks rocket launched from a platform 35 meters above the ground can be approimated b h = -5 t + 30t + 35, where h is the height in meters and t is the time in seconds. Find the time it takes the rocket to reach the ground after it is launched. 8-6 Solving Quadratic Equations b Factoring Use the Zero Product Propert to solve each equation. 5. ( + 1) ( + 3) = 0 6. ( - 6) ( - 3) = 0 7. ( + 6) ( - 3) = 0 8. ( + 7) ( - 10) = 0 Solve each quadratic equation b factoring. 9. - - 3 = 0 10. - 8 + 15 = 0 11. + = 6 1. -8-33 = - 8-7 Solving Quadratic Equations b Using Square Roots Solve using square roots. 13. 3 = 8 1. 36-9 = 0 15. -1 = - 1 8-8 Completing the Square Solve b completing the square. 16. + = 3 17. - 5 = 18. + 7 = 8 19. The width of a rectangle is feet shorter than its length. The area of the rectangle is square feet. Find the length and width. Round our answer to the nearest tenth of a foot. 8-9 Using the Quadratic Formula and the Discriminant Solve using the Quadratic Formula. Round our answer to the nearest hundredth. 0. + 5 + 1 = 0 1. 3 + 1 =. 5 + 8 = 3 Find the number of real solutions of each equation using the discriminant. 3. - 3 + = 0. + 1 + = 0 5. - 5 + = 0 8-10 Nonlinear Sstems 6. Solve the sstem b graphing. Check our answer. { = - 7 = - 7 7. Solve the sstem b substitution. Check our answer. { = + + 7 = + 5 Read to Go On? 599
EXTENSION Cubic Functions and Equations CC.9-1.F.IF.7c Graph polnomial functions, identifing zeros when suitable factorizations are available, and showing end behavior. Also CC.9-1.A.REI.10, CC.9-1.A.REI.11*, CC.9-1.A.APR.3 Objectives Recognize and graph cubic functions. Solve cubic equations. Vocabular cubic function cubic equation A cubic function is a function that can be written in the form f () = a 3 + b + c + d, where a 0. The parent cubic function is f () = 3. To graph this function, choose several values of and find ordered pairs. f() 8 1 1 0 0-1 -1 - -8 10 5 - - 0-5 -10 From the graph of f() = 3, ou can see the general shape of a cubic function. that the domain and the range are all real numbers. that the -intercept and the -intercept are both 0. A 6 - - 0 B The graph of f () = 3 + 5 + 1 illustrates another characteristic of the graphs of cubic functions. Points A and B are called turning points. In general, the graph of a cubic function will have two turning points. EXAMPLE 1 Graphing Cubic Functions Graph f () = 3 + 3 +. Identif the intercepts and give the domain and range. f() = 3 + 3 + f() 1 ( 1) 3 + 3 ( 1) 1 0 0 (0) 3 + 3 (0) + 0 1 (1) 3 + 3 (1) + 1 () 3 + 3 () + 6 Choose positive, negative, and zero values for, and find ordered pairs. 600 Chapter 8 Quadratic Functions and Equations
- - 0 - - -6 Plot the ordered pairs and connect them with a smooth curve. Notice that, in general, this graph falls from left to right. This is because the value of a is negative. The -intercept is 1. The -intercept is. The domain and range are all real numbers. Graph each cubic function. Identif the intercepts and give the domain and range. 1a. f () = ( 1) 3 1b. f () = 3 1 + 18 Previousl, ou saw that ever quadratic function has a related quadratic equation. Cubic functions also have related cubic equations. A cubic equation is an equation that can be written in the form a 3 + b + c + d = 0, where a 0. One wa to solve a cubic equation is b graphing the related function and finding its zeros. EXAMPLE Solving Cubic Equations b Graphing Solve 3 = b graphing. Check our answer. Step 1 Rewrite the equation in the form a 3 + b + c + d = 0. 3 = 3 + = 0 Add to both sides of the equation. Step Write and graph the related function: f () = 3 + f() = 3 - - + f() 1 ( 1) 3 - ( 1) - (-1) + 0 0 (0) 3 - (0) 0 + 6 1 (1) 3 - (1) - 1 + 0 () 3 - () - + 0 3 (3) 3 - (3) - 3 + 8 - - 0 Step 3 Find the zeros. The zeros appear to be 1, 1, and. Check these values in the original equation. 3 = 3 = 3 = ( 1) 3 ( 1) ( 1) 1 3 (1) 1 3 () ( 1) + 1 1 1 8 Etension 601
Solve each equation b graphing. Check our answer. a. 3 5 = 50 b. 3 + 1 = 30 + 00 Cubic equations can also be solved algebraicall. Man of the methods used to solve quadratic equations can be applied to cubic equations as well. EXAMPLE 3 Solving Cubic Equations Algebraicall Solve each equation. Check our answer. A ( + 5) 3 = 7 3 ( + 5) 3 = 3 7 + 5 = 3 = Check ( + 5) 3 = 7 ( + 5) 3 7 3 3 7 7 7 Take the cube root of both sides. Subtract 5 from both sides. Substitute for in the original equation. B 3 + 3 = 3 + 3 + = 0 ( + 3 + ) = 0 ( + 1)( + ) = 0 = 0 or + 1 = 0 or + = 0 = 1 or = The solutions are 0, 1, and. Add to both sides. Factor out on the left side. Factor the quadratic trinomial. Zero Product Propert Solve each equation. The factored epression must equal zero to use the Zero Product Propert. Check 3 + 3 = 3 + 3 = 3 + 3 = 0 3 + 3 (0) (0) (-1) 3 + 3 (-1) (-1) (-) 3 + 3 (-) (-) 0 + 0 0 1 + 3(1) 8 + 3() 0 0 1 + 3 8 + 1 C 3-3.15 = 1.5 3 + 1. 5-3.15 = 0 ( + 1.5-3.15) = 0 = 0 or + 1.5 3.15 = 0 = -1.5 ± (1.5) - (1)(-3.15) (1) -1.5 ± 3.75 = =.5 or = 1.5 Add 1.5 to both sides. Factor out on the left side. Zero Product Propert Quadratic Formula Simplif. The solutions are.5, 0, and 1.5. 60 Chapter 8 Quadratic Functions and Equations
Check Use a graphing calculator. Graph the related function and look for the zeros. The solutions look reasonable. Solve each equation. Check our answer. 3a. ( + ) 3 = 6 3b. 3 1 + = 0 3c. 3 + 3 = 10 EXTENSION Eercises Graph each cubic function. Identif the intercepts and give the domain and range. 1. f() = 3 + 3 + 6. g() = 3 + Solve each equation b graphing. Check our answer. 3. 3 6 =. 3 3 + 1 + 1 = 8 Solve each equation. Check our answer. 5. ( 9) 3 = 6 6. 8 + = 3 7. 5 3 + 3 = 8. The Send-It Store uses shipping labels that are in. tall and in. wide. Si labels fit on the front of the store s standard shipping bo with an area of 3 in left over. Three labels fit on the side of the bo. The volume of the bo is 108 in 3. What is the area of one label? 9. a. Graph the functions f() = 3, f() = 3 + 1, and f () = 3 + on the same coordinate plane. Describe an patterns ou observe. Predict the shape of the graph of f() = 3 + c. b. Graph the functions g() = 3, g() = ( 1) 3, and g() = ( ) 3 on the same coordinate plane. Describe an patterns ou observe. Predict the shape of the graph of g() = ( c) 3. 1 in. Use a graphing calculator to find the approimate solution(s) of each cubic equation. Round to the nearest hundredth. 10. 100 3 0 = 6 11. 3 5 + = 3 1. 1.3 3 3.65 = 0.3 13. 3_ 5 3 + - _ 1 = 9 1. Critical Thinking How man zeros can a cubic function have? What does this tell ou about the number of real solutions possible for a cubic equation? Etension 603
CHAPTER Vocabular ais of smmetr completing the square discriminant maimum minimum nonlinear sstem of equations parabola quadratic equation quadratic function verte zero of a function Complete the sentences below with vocabular words from the list above. 1. The? is the highest or lowest point on a parabola.. A? can also be called an -intercept of the function. 8-1 Identifing Quadratic Functions EXAMPLE Use a table of values to graph = -5 + 0. Step 1 Make a table of values. Choose values of and use them to find values of. 0 1 3 6 7 8 0 35 75 80 60 35 0 Step Plot the points and connect them. EXERCISES Tell whether each function is quadratic. Eplain. 3. = + 9-5. = - + 3 5. = - _ 1 6. = 5 3 + 8 Use a table of values to graph each quadratic function. 7. = 6 8. = - 9. = 1 _ 10. = -3 Tell whether the graph of each function opens upward or downward. Eplain. 11. = 5-1 1. = - + 3-7 8- Characteristics of Quadratic Functions EXAMPLE EXERCISES Find the zeros of = - - 6 from its graph. - 0 Use the graph to find the zeros. The zeros are -1 and 3. -8 Find the zeros of each quadratic function from its graph. Check our answer. 13. = + 3-10 1. = - - 60 Chapter 8 Quadratic Functions and Equations
8-3 Graphing Quadratic Functions EXAMPLE Graph = - 8-10. Step 1 Find the ais of smmetr. = _ -b a = _ -(-8) () = 8 _ = The ais of smmetr is =. Step 3 Find the -intercept. c = -10 Step Find the verte. = - 8-10 = () - 8 () - 10 = -18 The verte is (, -18). Step Find one more point on the graph. = (-1) - 8 (-1) -10 = 0 Let = -1. Use (-1, 0). Step 5 Graph the ais of smmetr and the points. Reflect the points and connect with a smooth curve. EXERCISES Graph each quadratic function. 15. = + 6 + 6 16. = - - 1 17. = - 8 + 7 18. = - 6-8 19. 3 + 6 = - 3 0. - + = 8-10 1. Water that is spraed upward from a sprinkler with an initial velocit of 0 m/s can be approimated b the function = -5 + 0, where is the height of a drop of water seconds after it is released. Graph this function. Find the time it takes a drop of water to reach its maimum height, the water s maimum height, and the time it takes the water to reach the ground. 8- Transforming Quadratic Functions EXAMPLE Compare the graph of g () = 3 - with the graph of f () =. Use the functions. Both graphs open upward because a > 0. The ais of smmetr is the same, = 0, because b = 0 in both functions. The graph of g ( ) is narrower than the graph of f () because 3 > 1. The verte of f () is (0, 0). The verte of g () is translated units down to (0, -). f () has one zero at the origin. g ( ) has two zeros because the verte is below the origin and the parabola opens upward. EXERCISES Compare the widths of the graphs of the given quadratic functions. Order functions with different widths from narrowest graph to widest.. f () =, g () = 3. f () = 6, g () = -6. f () =, g () = 1 _ 3, h () = 3 Compare the graph of each function with the graph of f () =. 5. g () = + 5 6. g () = 3-1 7. g () = + 3 Stud Guide: Review 605
8-5 Solving Quadratic Equations b Graphing EXAMPLE Solve - = - 8 b graphing the related function. Step 1 Write the equation in standard form. 0 = - 8 + Step Graph the related function. = - 8 + Step 3 Find the zeros. The onl zero is 1. The solution is = 1. EXERCISES Solve each equation b graphing the related function. 8. 0 = + + 3 9. 0 = + 6 + 9 30. - = 3 31. + 5 = 6 3. - = 6-3 33. 9 = 9 3. -3 + = 5 8-6 Solving Quadratic Equations b Factoring EXAMPLE Solve 3-6 = b factoring. 3-6 = Write the equation in 3-6 - = 0 standard form. 3 ( - - 8) = 0 Factor out 3. 3 ( + ) ( - ) = 0 Factor the trinomial. Zero Product 3 0, + = 0 or - = 0 Propert = - or = Solve each equation. EXERCISES Solve each quadratic equation b factoring. 35. + 6 + 5 = 0 36. + 9 + 1 = 0 37. - - 15 = 0 38. - - = 0 39. + 10 + 5 = 0 0. - 36 = -81 1. A rectangle is feet longer than it is wide. The area of the rectangle is 8 square feet. Write and solve an equation that can be used to find the width of the rectangle. 8-7 Solving Quadratic Equations b Using Square Roots EXAMPLE Solve = 98 using square roots. = 98 Divide both sides of the equation b to isolate. = 9 = ± 9 Take the square root of both sides. = ±7 Use ± to show both roots. The solutions are -7 and 7. EXERCISES Solve using square roots.. 5 = 30 3. - + 1 = 0. = -16 5. + 7 = 7 6. = 50 7. = 5 8. A rectangle is twice as long as it is wide. The area of the rectangle is 3 square feet. Find the rectangle s width. 606 Chapter 8 Quadratic Functions and Equations
8-8 Completing the Square EXAMPLE Solve - 6 = -5 b completing the square. _ ( -6 ) = 9 Find ( b ). - 6 + 9 =-5 + 9-6 + 9 = Complete the square. ( - 3) = Factor the trinomial. 3 =± Take the square root of both sides. - 3 = ± Use the ± smbol. - 3 = or - 3 =- Solve each = 5 or = 1 equation. The solutions are 5 and 1. EXERCISES Solve b completing the square. 9. + = 8 50. + = 1 51. - 1 + 10 = 0 5. - 10 =-0 53. A homeowner is planning an addition to her house. She wants the new famil room to be a rectangle with an area of 19 square feet. The contractor sas that the length needs to be more feet than the width. What will the dimensions of the new room be? 8-9 The Quadratic Formula and the Discriminant EXAMPLE Solve + + = 0 using the Quadratic Formula. = -b ± b - ac a - ± - (1)() = (1) = - ± 16-16 = _ - ± 0 = _ - = - The solution is = -. Write the Quadratic Formula. Substitute for a, b, and c. Simplif. EXERCISES Solve using the Quadratic Formula. 5. - 5-6 = 0 55. - 9-5 = 0 56. - 8 + = 0 57. - 6 = -7 Find the number of real solutions of each equation using the discriminant. 58. - 1 + 36 = 0 59. 3 + 5 = 0 60. - 13 = -0 61. 6-0 = 15 + 1 8-10 Nonlinear Sstems EXAMPLE Solve { = b substitution. = + + =. 0 = - - Substitute + for in the first equation. 0 = ( - )( + 1) Factor the trinomial. - = 0 or + 1 = 0 Solve the equations. = or = -1 Write one of the = + = + original equations. = + =-1 + Substitute each -value = = 1 and solve for. The solutions are (, ) and (-1, 1). EXERCISES Solve each sstem. 6. { = - = - - 1 63. { = - 3-1 = - 5 6. { = + 5 + =- - 6 Stud Guide: Review 607
CHAPTER Tell whether each function is quadratic. Eplain. 1. { (10, 50), (11, 71), (1, 9), (13, 119), (1, 16) }. 3 + = + 3 3. Tell whether the graph of = - + 7-5 opens upward or downward and whether the parabola has a maimum or a minimum.. Estimate the zeros of the quadratic function. 5. Find the ais of smmetr of the parabola. 6. Find the verte of the graph of = + 6 + 8. 7. Graph the quadratic function = - +. Compare the graph of each function with the graph of f () =. 8. g () = - - 9. h () = 1 _ 3 + 1 10. g () = 3-11. A hammer is dropped from a 0-foot scaffold. Another one is dropped from a 60-foot scaffold. a. Write the two height functions and compare their graphs. Use h (t) = -16 t + c, where c is the height of the scaffold. b. Use the graphs to estimate when each hammer will reach the ground. 1. A rocket is launched with an initial vertical velocit of 110 m/s. The height of the rocket in meters is approimated b the quadratic equation h = -5 t + 110t where t is the time after launch in seconds. About how long does it take for the rocket to return to the ground? Solve each quadratic equation b factoring. 13. + 6 + 5 = 0 1. - 1 = -36 15. - 81 = 0 Solve b using square roots. 16. - = -7 17. 9-9 = 0 18. 3 + 1 = 0 Solve b completing the square. 19. + 10 = -1 0. - 6 + = 0 1. + 16 = 0 Solve each quadratic equation. Round to the nearest hundredth if necessar.. + 3-0 = 0 3. + 7 = -5. 8 + 3-1 = 0 Find the number of real solutions of each equation using the discriminant. 5. - + 1 = 0 6. + 5-5 = 0 7. 1 _ + 8 = 0 8. Solve the sstem. { = + 7 + 6 = + 1 608 Chapter 8 Quadratic Functions and Equations
CHAPTER FOCUS ON SAT SUBJECT TESTS In addition to the SAT, some colleges require the SAT Subject Tests for admission. Colleges that don t require the SAT Subject Tests ma still use the scores to learn about our academic background and to place ou in the appropriate college math class. You ma want to time ourself as ou take this practice test. It should take ou about 6 minutes to complete. Take the SAT Subject Test in mathematics while the material is still fresh in our mind. You are not epected to be familiar with all of the test content, but ou should have completed at least three ears of college-prep math. 1. The graph below corresponds to which of the following quadratic functions? (A) f () = + - 5 (B) f () = - - + 3 (C) f () = - + 5 - (D) f () = - - + 5 (E) f () = - - 3 + 5. What is the sum of the solutions to the equation 9-6 = 8? (A) _ 3 (B) _ 3 (C) 1 _ 3 (D) - _ 3 (E) - 8 _ 3 3. If h () = a + b + c, where b - ac < 0 and a < 0, which of the following statements must be true? I. The graph of h () has no points in the first or second quadrants. II. The graph of h () has no points in the third or fourth quadrants. III. The graph of h () has points in all quadrants. (A) I onl (B) II onl (C) III onl (D) I and II onl (E) None of the statements are true.. What is the ais of smmetr for the graph of a quadratic function whose zeros are - and? (A) = - (B) = 0 (C) = 1 (D) = (E) = 6 5. How man real-number solutions does 0 = - 7 + 1 have? (A) None (B) One (C) Two (D) All real numbers (E) It is impossible to determine. College Entrance Eam Practice 609
CHAPTER Etended Response: Eplain Your Reasoning Etended response test items often include multipart questions that evaluate our understanding of a math concept. To receive full credit, ou must answer the problem correctl, show all of our work, and eplain our reasoning. Use complete sentences and show our problem-solving method clearl. Etended Response Given 1 + = - 3 and = - 1, identif which is a quadratic function. Provide an eplanation for our decision. For the quadratic function, tell whether the graph of the function opens upward or downward and whether the parabola has a maimum or a minimum. Eplain our reasoning. Read the solutions provided b two different students. Student A Ecellent eplanation The response includes the correct answers along with a detailed eplanation for each part of the problem. The eplanation is written using complete sentences and is presented in an order that is eas to follow and to understand. It is obvious that this student knows how to determine and interpret a quadratic function. Student B Poor eplanation The response includes the correct answers, but the eplanation does not include details. The reason for defining the function as quadratic does not show knowledge of the concept. The student shows a lack of understanding of how to write and interpret a quadratic function in standard form. 610 Chapter 8 Quadratic Functions and Equations
Include as man details as possible to support our reasoning. This increases the chance of getting full credit for our response. Read each test item and answer the questions that follow. Item A The height in feet of a tennis ball seconds after it is ejected from a serving machine is given b the ordered pairs { (0, 10), (0.5, 9), (1, 7), (1.5, ), (, 0) }. Determine whether the function is quadratic. Find its domain and range. Eplain our answers. 1. What should a student include in the eplanation to receive full credit?. Read the two eplanations below. Which eplanation is better? Wh? Item C A science teacher set off a bottle rocket as part of a lab eperiment. The function h = -16 t + 96t represents the height in feet of a rocket that is shot out of a bottle with an initial vertical velocit of 96 feet per second. Find the time that the rocket is in the air. Eplain how ou found our answer.. Read the two responses below. a. Which student provided the better eplanation? Wh? b. What advice would ou give the other student to improve his or her eplanation? Student C Student A Student B Student D Item B The height of a golf ball can be approimated b the function = -5 + 0 + 8, where is the height in meters above the ground and is the time in seconds after the ball is hit. What is the maimum height of the ball? How long does it take for the ball to reach its maimum height? Eplain. 3. A student correctl found the following answers. Use this information to write a clear and concise eplanation. Item D The base of a parallelogram is 1 centimeters more than its height. The area of the parallelogram is 13 square centimeters. Eplain how to determine the height and base of the figure. What is the height? What is the base? 5. Read the following response. Identif an areas that need improvement. Rewrite the response so that it will receive full credit. Test Tackler 611
CHAPTER State Test Practice CUMULATIVE ASSESSMENT Multiple Choice 1. Which epression is NOT equal to the other three? 0 1 1 0 1 1 (-1) 0. Which function s graph is a translation of the graph of f () = 3 + seven units down? f () = - + f () = 10 + f () = 3-3 f () = 3 + 11 3. The area of a circle in square units is π (9 + + 9). Which epression represents the circumference of the circle in units? π (3 + 7) π (3 + 7) π (3 + 7) 6 + 1. CberCafe charges a computer station rental fee of $5, plus $0.0 for each quarter-hour spent surfing. Which epression represents the total amount Carl will pa to use a computer station for three and a half hours? 5 + 0.0 (3.5) 5 + 0.0 (3.5) () 5 + 0.0 _ 3.5 5 + _ 1 _ 0.0 3.5 5. What is the numerical solution to the equation five less than three times a number equals four more than eight times the number? - 9 _ 5-1 _ 5 1_ 11 1_ 5 6. Which is a possible situation for the graph? A car travels at a stead speed, slows down in a school zone, and then resumes its previous speed. A child climbs the ladder of a slide and then slides down. A person flies in an airplane for a while, parachutes out, and gets stuck in a tree. The number of visitors increases in the summer, declines in the fall, and levels off in the winter. 7. Which of the following is the graph of f () = - +? -3-3 3 0 3-3 0 3-3 0 3-3 3-3 -6 8. The value of varies directl with, and = 0 when = -5. Find when = 8. 5-8 -1-6 9. What is the slope of the line that passes through the points (, 7) and (5, 3)? -_ 1 1_ - 0 3 61 Chapter 8 Quadratic Functions and Equations
10. Putting Green Mini Golf charges a $ golf club rental fee plus $1.5 per game. Good Times Golf charges a $1.5 golf club rental fee plus $3.75 per game. Which sstem of equations could be solved to determine for how man games the cost is the same at both places? = + 1.5 = 3.75 + 1.5 = - 1.5 = -3.75 + 1.5 = 1.5 + = 3.75 + 1.5 = 1.5 - = 1.5 + 3.75 11. The graph of which quadratic function has an ais of smmetr of = -? = - + 3 = + + 3 = - + 3 = + + 3 1. Which polnomial is the product of - and - + 1? - + 17-3 - 8 + 17-3 + 17-3 - 15 + Gridded Response The problems on man standardized tests are ordered from least to most difficult, but all items are usuall worth the same amount of points. If ou are stuck on a question near the end of the test, our time ma be better spent rechecking our answers to earlier questions. 13. The length of a rectangle is units greater than the width. The area of the rectangle is square units. What is its width in units? 1. Find the value of the discriminant of the equation 0 = - + 3 +. Short Response 16. The data in the table shows ordered pair solutions to a linear function. Find the missing -value. Show our work. 17. Answer the following questions using the function f () = + - 1. a. Make a table of values and give five points on the graph. b. Find the ais of smmetr and verte. Show all calculations. 18. a. Show how to solve - - 8 = 0 b graphing the related function. Show all our work. b. Show another wa to solve the equation in part a. Show all our work. 19. What can ou sa about the value of a if the graph of = a - 8 has no -intercepts? Eplain. Etended Response 0. The graph shows the quadratic function f () = a + b + c. a. What are the solutions of the equation 0 = a + b + c? Eplain how ou know. - 0 - - b. If the point (-5, 1) lies on the graph of f (), the point (a, 1) also lies on the graph. Find the value of a. c. What do ou know about the relationship between the values of a and b? Use the coordinates of the verte in our eplanation. d. Use what ou know about solving quadratic equations b factoring to make a conjecture about the values of a, b, and c in the function f () = a + b + c. - -7-1 -3 0 1 5 9 15. What is the positive solution of = 10 +? Round our answer to the nearest hundredth if necessar. Standardized Test Prep 613