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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. Warm Ups Factor each quadratic. Wednesday, October 21, 2015

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 2 Wednesday, October 21, 2015 Essential Question: How do we find the roots or zeros of quadratic functions of the form y = ax 2 + bx +c ? Lesson 3.4B

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 3 Solutions of a quadratic equation or function. If y = f (x) is a quadratic function and a is a real number then the following statements are equivalent. 1. x = a is a zero of f. 2. x = a is a root of f. 3. x = a is a solution of the quadratic equation f (x) = 0. 4. (x – a) is a factor of the quadratic f (x). 5. (a, 0) is an x-intercept of the graph of y = f (x).

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 4 We must set the equation equal to zero before we factor. 1. +3 - 4 +4 The solutions are and.

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 5 We must set the equation equal to zero before we factor. 2. +21 +90 +15+6 The solutions are and.

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 6 We must substitute zero for “y” and solve the quadratic equation. 3. +3 - 28 +7-4 The zeros are and.

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 7 4. -15 +36 -12-3 The roots are and. We must substitute zero for “y” and solve the quadratic equation.

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 8 5. +4 +2 The x-intercept is. We must substitute zero for “y” and solve the quadratic equation. If you solve the other equation you will get the same solution. We have only one unique solution!

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 9 6. -7 +6 -6 The x-intercepts are and. We must substitute zero for “y” and solve the quadratic equation.

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. A. B. Which of the following is the graph of the function f(x) = (x + 3)(x – 3) ? C. 7.

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. Find the x-intercepts 8.

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. Find the zeroes of the function 9.

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 10.

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. Find the maximum value of a quadratic function The height y (in feet) of a baseball t seconds after it is hit is given by this function: Baseball y = –16t 2 + 96t + 3 Find the maximum height of the baseball. SOLUTION The maximum height of the baseball is the y -coordinate of the vertex of the parabola with the given equation. 7.

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. Find the maximum value of a quadratic function y = – 16t 2 + 96t +3 Write original function. The verte x is (3, 147), so the maximum height of the baseball is 147 feet. ANSWER

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. SOLUTION The maximum height of the baseball is the y -coordinate of the vertex of the parabola with the given equation. y = – 16t 2 + 80t +2 Write original function. Suppose the height of the baseball is given by y = – 16t 2 + 80t + 2. Find the maximum height of the baseball. Find the maximum value of a quadratic function The vertex is (2.5, 102), so the maximum height of the baseball is 102 feet. 8.

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MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 17 THE END Homework page 81 # 1 – 3 all, 16 – 21 all.

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