York UniversityMATHEMATIC 3410RootsQuadraticSE

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1. The roots of a quadratic equation are the values of x that make the related function zero. The real roots are also the x-intercepts of the parabola. Look at the graph of y = x 2 – 4x + 3. A. ... How many roots does x 2 – 4x + 3 = 0 have? What are the roots? B. Change c to 4.0. How many roots does x 2 – 4x + 4 = 0 have? 2. Now graph y = x 2 – 4x + 8 in the Gizmo, and look at the resulting parabola. Do you think x 2 – 4x + 8 = 0 has any real roots? Explain. 3. Vary the values of a, b, and c. In general, how many real roots are possible for a quadratic equation? 4. Graph y = x 2 + 6x + 5. Turn on Show axis of symmetry x = –b/(2a). The axis of symmetry is a line that divides a parabola into two halves that are mirror images. A. How does the location of the axis of symmetry relate to the location of the two x-intercepts? B. Move the a, b, and c sliders. Which values affect the axis of symmetry? C. The equation of the axis of symmetry is x = a b 2  . How does this explain what you observed above? 5. Suppose you know the line of symmetry for a quadratic function. A. From just this information, can you find the x-intercepts? Explain. B. Suppose the axis of symmetry of the graph of a quadratic function is at x = 6. If one root of the related quadratic equation is –1.5, what is the other root? This study source was downloaded by 100000827637563 from CourseHero.com on 09-12-2021 15:37:11 GMT -05:00 https://www.coursehero.com/file/73552774/RootsQuadraticSEpdf/ This study resource was shared via CourseHero.com 2019 Activity B: The quadratic formula Get the Gizmo ready:  Be sure the CONTROLS and REAL PLANE tabs are selected. 1. Some quadratic equations are difficult to factor. In these cases, you can use the quadratic formula, x = a b b ac 2 4 2    , to find the roots of the quadratic equation ax2 + bx + c = 0. A. Graph y = 3x 2 – x – 4. Select the SOLUTION tab to see how the quadratic formula is used to find the roots of 3x 2 – x – 4 = 0. What are the roots? Click on the CONTROLS tab to check that these are the x-intercepts of the graph. B. Use the quadratic formula to find the roots of 2x 2 + x – 10 = 0. Show your work in the space to the right. Then check your answer in the Gizmo. 2. The discriminant is the part of the quadratic formula that is under the radical, b 2 – 4ac. It provides useful information about the number of real roots of a quadratic equation. On the CONTROLS tab, turn on Show discriminant computation. A. Graph each quadratic function listed to the right in the Gizmo. Then state the number of real roots of the related equation (ax2 + bx + c = 0) and give the discriminant. B. Vary a, b, and c, and watch how the number of real roots and the discriminant change. In general, how does the discriminant relate to the number of real roots? C. Why do you think the discriminant determines the number of real roots of a quadratic equation? (Activity B continued on next page) Function Number of real roots Discriminant y = x 2 + 6x + 9 y = x 2 – 5x – 8 y = x 2 – 4x + 6 This study source was downloaded by 100000827637563 from CourseHero.com on 09-12-2021 15:37:11 GMT -05:00 https://www.coursehero.com/file/73552774/RootsQuadraticSEpdf/ This study resource was shared via CourseHero.com 2019 Activity B (continued from previous page) 3. A fraction with a sum in the numerator can be rewritten as the sum of two fractions with the same denominator. For example, 2 1 x = 2 1 + 2 x . A. In the space to the right, rewrite the quadratic formula as the sum of two fractions. B. What line does the first fraction describe? C. What is the second numerator the same as? D. Whenever a quadratic function has two x-intercepts, they are always equidistant from the axis of symmetry. Use your findings from above to explain why this is true. [Show More]

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