Determine a Quadratic Equation Given Its Roots - McGraw-Hill ...

14. Chapter Problem The actuarial firm whereAndrea has her co-op placement wassent a set of data that follows a quadraticfunction. The data supplied comparedthe number of years of driving experiencewith the number of collisions reported toan insurance company in the last month.Andrea was asked to recover the data lostwhen the paper jammed in the fax machine.Only three data points can be read. Theyare (5, 22), (8, 28), and (9, 22). The values off (x) for x 5 6 and x 5 7 are missing. Andreadecided to subtract the y-value of 22 fromeach point so that she would have twozeros: (5, 0), (8, 6), and (9, 0).a) Use these three points to find aquadratic function that can be used tomodel the adjusted data.b) Add a y-value of 22 to this function fora quadratic function that models theoriginal data.c) Use this function to find the missingvalues for x 5 6 and x 5 7.15. An arch of aReasoning and Provinghighway overpassis in the shape of aparabola. The archRepresentingProblem SolvingSelecting Toolsspans a distanceof 12 m from oneside of the road toConnectingReflectingCommunicatingthe other. The height of the arch is 8 m ata horizontal distance of 2 m from each sideof the arch.a) Sketch the quadratic function if thevertex of the parabola is on the y-axisand the road is along the x-axis.b) Use this information to determine thefunction that models the arch.c) Find the maximum height of the arch tothe nearest tenth of a metre.16. Use the information from question 15, butinstead of having the vertex on the y-axis,put one side of the archway at the origin ofthe grid. You will get a different equationbecause the zeros are now at 0 and 12,rather than at 6 and 6.a) Find the equation of the quadraticfunction for this position.b) Find the maximum height of theoverpass and compare the result to theheight calculated in question 15.17. Explain how the two equations developedin questions 15 and 16 can model the samearch, even though the equations are different.Achievement Check18. A quadratic function has zeros 2 and 6and passes through the point (3, 15).a) Find the equation of the quadraticfunction in factored form.b) Write the function in standard form.c) Complete the square to convert thestandard form to vertex form, and statethe vertex.d) Use partial factoring to verify youranswer to part c).e) Find a second quadratic function withthe same zeros as in part a), but passingthrough the point (3, 30). Express thefunction in standard form.f) Graph both functions. Explain how thegraphs can be used to verify that theequations in parts a) and e) are correct.C Extend19. Is it possible to determine the definingequation of a function given the followinginformation? If so, justify your answer andprovide an example.a) the vertex and one x-interceptb) the vertex and one other point on theparabolac) any three points on the parabola20. Math Contest Determine an equation fora quadratic function __ with zeros atx 5 ​__1 ​ 7 ​​.322. Math Contest Show that the graph off (x) 5 ax 2 c has no x-intercept if ac 0.1.6 Determine a Quadratic Equation Given Its Roots • MHR 59