1.6. Determine a Quadratic Equation Given Its Roots. Investigate


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1 1.6 Determine a Quadratic Equation Given Its Roots Bridges like the one shown often have supports in the shape of parabolas. If the anchors at either side of the bridge are 4 m apart and the maximum height of the support is 6 m, what function models the parabolic curve of the support? Engineers need to determine this function to ensure that the bridge is built to proper specifications. How can the given data be used to model the equation of the parabola? Tools grid paper Investigate How can ou connect the zeros to a form of the quadratic function? In the introduction, information was given about a parabolic support under a bridge. What equation will model the parabolic curve of the support if the vertex is on the axis and the points of attachment of the supports are on the xaxis? Method 1: Use Pencil and Paper 1. Use the information given to identif three points: the two xintercepts and the vertex. Sketch the function. Label the three known points.. The intercept form of a quadratic function is 5 a(x r)(x s), where r and s are the xintercepts. Write the function in this form using the data from the original problem for the xintercepts. 3. How can ou use the third known point to find the value of a? 4. a) Write a function in factored form for the bridge support. b) Express the function from part a) in standard form. 5. Reflect Can ou write the equation of a quadratic function given its zeros? If so, describe how. If not, explain wh not. 5 MHR Functions 11 Chapter 1
2 Method : Use a Graphing Calculator 1. Use the information given to identif three points and draw a sketch. Enter the three data points into a graphing calculator using L1 and L. Tools graphing calculator. Use quadratic regression to find the equation of the quadratic function in the form = ax bx c. 3. Enter the function for Y1 and graph the equation to verif. 4. a) The intercept form of a quadratic function is 5 a(x r)(x s), where r and s are the xintercepts. Write this form of the function, using the same value of a as found in step and the data from the original problem. Enter this form of the function as Y on the graphing calculator, choose a different thickness for the new line, and graph this line. b) What do ou notice occurs on the displa of the graphing calculator as the second parabola is graphed? 5. Reflect Can ou determine the equation of a quadratic function given its zeros? If so, describe how. If not, explain wh not. Example 1 Find the Equation of a Famil of Quadratic Functions Find the equation, in factored form, for a famil of quadratic functions with the given xintercepts. Sketch each famil, showing at least three members. a) 4 and b) 0 and 5 c) 3 and 3 d) 6 is the onl xintercept Connections Functions that have a common propert are called a famil. In grade 9, ou worked with families of linear functions that have the same slope: the are parallel lines. Solution a) Since x 5 4 and x 5 are roots of the equation, x 4 and x are factors of the function. The equation for this famil is f (x) 5 a(x 4)(x ). 4 f(x) = (x 4)(x ) x 4 f(x) = (x 4)(x ) f(x) = 4(x 4)(x ) 1.6 Determine a Quadratic Equation Given Its Roots MHR 53
3 b) Since x 5 0 and x 5 5 are roots of the equation, x and x 5 are factors of the function. The equation for this famil is f (x) 5 ax(x 5) f(x) = x(x + 5) x f(x) = x(x + 5) f(x) = x(x + 5) c) Since 3 and 3 are the xintercepts, x 3 and x 3 are factors. The equation for this famil is f (x) 5 a(x 3)(x 3). 6 4 f(x) = (x 3)(x + 3) f(x) = (x 3)(x + 3) x f(x) = 1 (x 3)(x + 3) 4 6 d) Since x 5 6 is the onl zero, x 6 must be a repeated factor. The equation for this famil is f (x) 5 a(x 6). 4 f(x) = (x 6) f(x) = (x 6) x f(x) = 1 (x 6) f(x) = (x 6) 4 Example Determine the Exact Equation of a Quadratic Function Find the equation of the quadratic function with the given zeros and containing the given point. Express our answers in standard form. a) and, containing the point (0, 3) b) double zero at x 5, containing the point (3, 10) c) 3 5 and 3 5, containing the point (, 1) 54 MHR Functions 11 Chapter 1
4 Solution a) Since and 3 are zeros, then x and x 3 are factors. f (x) 5 a(x )(x 3) Substitute the given point: f (0) a(0 )(0 3) 3 5 6a a 5 1_ The function, in factored form, is f (x) 5 1_ (x )(x 3). In standard form: f (x) 5 1_ (x x 6) 5 1_ x 1_ x 3 Check b graphing the function using a graphing calculator. b) Since is a double zero, the factor x is repeated. f (x) 5 a(x ) Substitute the point: f (3) a(3 ) a a The function, in factored form, is f (x) 5 0.4(x ). In standard form: f (x) 5 0.4(x 4x 4) 5 0.4x 1.6x 1.6 c) Since 3 5 and 3 5 are zeros, then ( x ( 3 5 ) ) and ( x ( 3 5 ) ) are factors. f (x) 5 a ( x ( 3 5 ) ) ( x ( 3 5 ) ) 5 a ( x 3 5 ) ( x 3 5 ) This is in the form (c d)(c + d), where 5 ) ] c = x 3 and d = 5. 5 a [ (x 3) ( 5 a(x 6x 9 5) 5 a(x 6x 4) Substitute the point: f () a( 6() 4) 1 5 a( 4) a 5 3 The function, in factored form, is f (x) 5 3(x 6x 4). In standard form: f (x) 5 3(x 6x 4) 5 3x 18x 1 Simplif and solve for a. 1.6 Determine a Quadratic Equation Given Its Roots MHR 55
5 Example 3 Represent Given Information as a Quadratic Function Reasoning and Proving Representing Selecting Tools Problem Solving Connecting Reflecting Communicating The parabolic opening to a tunnel is 3 m wide measured from side to side along the ground. At the points that are 4 m from each side, the tunnel entrance is 6 m high. a) Sketch a diagram of the given information. b) Determine the equation of the function that models the opening to the tunnel. c) Find the maximum height of the tunnel, to the nearest tenth of a metre. Solution Connections If ou assume that one side of the tunnel is at the origin, ou will get a different form of the equation. It will be a translation of the one found here. You will examine the effects of translations on the equation of a function in Chapter. a) The point (1, 6) comes from the information given. You are told that 4 m from each side, the height is 6 m. The point ( 1, 6) can also be used, giving the same answer. b) Use the xintercepts 16 and 16. Write the general function, f (x) 5 a(x 16)(x 16). To solve for a, substitute the point (1, 6). 6 5 a(1 16)(1 16) 6 5 a( 4)(8) 6 5 a( 11) a 5 6_ _ 56 The function that models the opening to the tunnel is f (x) 5 3_ (x 16)(x 16). 56 c) The maximum height of the tunnel will occur halfwa between the two xintercepts. This means a value of x 5 0. f (0) 5 3_ (0 16)(0 16) _ 56 ( 16)(16) The maximum height of the tunnel is 13.7 m, to the nearest tenth of a metre (1, 6) x 56 MHR Functions 11 Chapter 1
6 Ke Concepts The zeros can be used to find the equation of a famil of quadratic functions with the same xintercepts. To determine an individual quadratic function, ou also need to be given one other point on the function. Communicate Your Understanding C1 Outline the steps needed to find the equation of a quadratic function given the xintercepts and one other point on the function. C C3 You are given an equation for a famil of quadratic functions with the same xintercepts. Rita sas, The vertex is the onl point that will not allow ou to determine the exact equation, as it is at the centre of the function, and more than one function can be found. Ronnie claims, The vertex is as good as an other point in finding the exact function. Who is correct? Explain. Mona has decided that if she is given a fraction such as 1_ as one of the xintercepts, she can use the binomial (x 1) instead of ( x 1_ ) and get the same quadratic function. Is she correct? Explain. A Practise For help with questions 1 and, refer to Example Determine the equation, in factored form, of a famil of quadratic functions with each pair of roots. Sketch a graph to show four graphs in each famil. a) x 5 3 and x 5 6 b) x 5 1 and x 5 1 c) x 5 3 and x 5 4. Express each equation in question 1 in standard form. For help with question 3 to 5, refer to Example. 3. Find the equation of a quadratic function that has the given xintercepts and contains the given point. Express each function in factored form. Graph each function to check. a) 3 and 5, point (4, 3) b) 4 and 7, point ( 3, 1) c) 0 and _, point ( 1, 5) 3 4. Write each function in question 3 in standard form. 5. Find the equation of the quadratic function that has the given zeros and contains the given point. Express each function in standard form. Graph each function to check. _ a) 1 11, point (4, 6) b) 7, point (1, ) c) 5, point (, 14) 6. Find the equation of the quadratic function that has the given zeros and contains the given point. Express each function in vertex form. Graph each function to check. a) 3 and 1, point (1, ) b) 1 and, point (0, 4) c) 3 and 5, point (1, 4) 1.6 Determine a Quadratic Equation Given Its Roots MHR 57
7 B Connect and Appl For help with question 7, refer to Example A soccer ball is kicked from the ground. After Representing travelling a horizontal distance of 35 m, it just passes over a Connecting 1.5mtall fence before hitting the ground 37 m from where it was kicked. Reasoning and Proving Problem Solving Communicating Selecting Tools Reflecting b) x (, 36) m 37 m 1.5 m a) Considering the ground to be the xaxis and the vertex to be on the axis, determine the equation of a quadratic function that can be used to model the parabolic path of the ball. b) Determine the maximum height of the ball. c) How far has the ball travelled horizontall to reach the maximum height? d) Develop a new equation for the quadratic function that represents the height of the ball, considering the ball to have been kicked from the origin. e) Outline the similarities and differences between the functions found in parts a) and d). f) Use Technolog Use a graphing calculator to compare the solutions. 8. Determine the equation in standard form for each quadratic function shown. a) (, 0) (3, 0) 0 4 x 4 (4, 3) c) x 0 (, 8) 9. Use Technolog For each part in question 8, use a graphing calculator to verif our solution b plotting the three points as well as entering the quadratic function. Explain how ou can use this method to check that our solution is correct. 10. Explain how the technique studied in this section can be used to find the equation for the quadratic function if the onl xintercept is the origin and ou are given one other point on the function. 11. Find the quadratic function that has onl one xintercept and passes through the given point. a) xintercept of 0, point (5, ) b) xintercept of 5, point (4, 3) c) xintercept of 1, point (, 6) 1. Use Technolog Verif our solutions to question 11 using a graphing calculator. 13. If the function f (x) 5 ax 5x c has onl one xintercept, what is the mathematical relationship between a and c? 58 MHR Functions 11 Chapter 1
8 14. Chapter Problem The actuarial firm where Andrea has her coop placement was sent a set of data that follows a quadratic function. The data supplied compared the number of ears of driving experience with the number of collisions reported to an insurance compan in the last month. Andrea was asked to recover the data lost when the paper jammed in the fax machine. Onl three data points can be read. The are (5, ), (8, 8), and (9, ). The values of f (x) for x 5 6 and x 5 7 are missing. Andrea decided to subtract the value of from each point so that she would have two zeros: (5, 0), (8, 6), and (9, 0). a) Use these three points to find a quadratic function that can be used to model the adjusted data. b) Add a value of to this function for a quadratic function that models the original data. c) Use this function to find the missing values for x 5 6 and x An arch of a Reasoning and Proving highwa overpass is in the shape of a parabola. The arch Representing Problem Solving Selecting Tools spans a distance of 1 m from one side of the road to Connecting Reflecting Communicating the other. The height of the arch is 8 m at a horizontal distance of m from each side of the arch. a) Sketch the quadratic function if the vertex of the parabola is on the axis and the road is along the xaxis. b) Use this information to determine the function that models the arch. c) Find the maximum height of the arch to the nearest tenth of a metre. 16. Use the information from question 15, but instead of having the vertex on the axis, put one side of the archwa at the origin of the grid. You will get a different equation because the zeros are now at 0 and 1, rather than at 6 and 6. a) Find the equation of the quadratic function for this position. b) Find the maximum height of the overpass and compare the result to the height calculated in question Explain how the two equations developed in questions 15 and 16 can model the same arch, even though the equations are different. Achievement Check 18. A quadratic function has zeros and 6 and passes through the point (3, 15). a) Find the equation of the quadratic function in factored form. b) Write the function in standard form. c) Complete the square to convert the standard form to vertex form, and state the vertex. d) Use partial factoring to verif our answer to part c). e) Find a second quadratic function with the same zeros as in part a), but passing through the point (3, 30). Express the function in standard form. f) Graph both functions. Explain how the graphs can be used to verif that the equations in parts a) and e) are correct. C Extend 19. Is it possible to determine the defining equation of a function given the following information? If so, justif our answer and provide an example. a) the vertex and one xintercept b) the vertex and one other point on the parabola c) an three points on the parabola 0. Math Contest Determine an equation for a quadratic function with zeros at x Math Contest Show that the graph of f (x) 5 ax c has no xintercept if ac Determine a Quadratic Equation Given Its Roots MHR 59
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