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3.2 Quadratic Functions & Graphs

3.2 Quadratic Functions & Graphs. Quiz. Write out the general form of a quadratic equation. f(x) = _____________. Quadratic Function. Complete the Square. Complete the Square. General Form f (x) = ax 2 + bx + c. Standard Form f (x) = a(x – h) 2 + k.

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3.2 Quadratic Functions & Graphs

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  1. 3.2 Quadratic Functions & Graphs

  2. Quiz • Write out the general form of a quadratic equation. f(x) = _____________

  3. Quadratic Function

  4. Complete the Square Complete the Square General Form f(x) = ax2 + bx + c Standard Form f(x) = a(x – h)2 + k x2 + 2px + p2 = (x + p)2 x2 - 2px + p2 = (x - p)2

  5. Complete the Square • Example: Given f(x) = 2x2 - 8x + 1,complete the square to put it into the form f(x) = a(x – h)2 + k. • How about f(x) = x2 - 3x + 1?

  6. The Graph of a Quadratic Maximum point Vertex y y x x Minimum point Vertex Axis of symmetry x = h Axis of symmetry x = h

  7. Find vertex of a parabola Transformation form: f(x) = a(x – h)2 + k Vertex : ( h, k ) Axis of symmetry: x = h Transformation form: f(x) = ax2 + bx + c Vertex : ( - b/2a, f(- b/2a) ) Axis of symmetry: x = -b/2a

  8. Graphing parabolas • Determine if the graph opens up or down • Determine the vertex ( h, k ) • Find the y – intercept • Plot the vertex and at least 2 additional points on one side of the vertex • Use symmetry finish the other half Example: f(x) = 2x2 + x - 3

  9. Application • Write the equation of the parabola with vertex at (8, 3) passing through (10, 5). f(x) = a ( x – h )2 + k 10 8 3 5

  10. Height of a Projected Object • If air resistance is neglected, the height s ( in feet ) of an object projected directly upward from an initial height s0 feet with initial velocity v0 feet per second is s (t) = -16t2 + v0t + s0, where t is the number of seconds after the object is projected.

  11. Application A ball is thrown directly upward from an initial height of 100 feet with an initial velocity of 80 feet per second. • Give the function that describes the height of the ball in terms of time t. • Graph this function so that the y-intercept, the positive x-intercept, and the vertex are visible. • If the point (4.8, 115.36) lies on the graph of the function. What does this mean for this particular situation? • After how many seconds does the projectile reach its maximum height? What is the maximum height? Solve analytically and graphically. • For what interval of time is the height of the ball greater than 160 feet? Determine the answer graphically. • After how many seconds will the ball fall to the ground? Determine the answer graphically.

  12. Homework • PG. 174: 3-48(M3), 17 instead of 18 PG. 175: 57 – 72 (M3) • Key: 6, 27, 36, 57, 63 • Reading: 3.3 Quadratic Equation & Ineq.

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