 # Essential Question(s): How can you tell if a quadratic function a) opens up or down b) has a minimum or maximum, and c) how many x-intercepts it has?

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Essential Question(s): How can you tell if a quadratic function a) opens up or down b) has a minimum or maximum, and c) how many x-intercepts it has?

 “Wait… didn’t we do this already?” I tried to warn you… The notes that follow in yellow, I will expect you to memorize (meaning: they won’t be given to you on a quiz)  Quadratic Functions are parabolas (‘U’ shaped) and a) Can open either upward or downward b) Always have a vertex which is either the maximum or minimum  Opening up == minimum, opening down == maximum c) Always have exactly one y-intercept d) Can have 0, 1, or 2 x-intercepts  The x-intercept(s) are the solution(s) [roots] of the equation

 Quadratic Functions can be written in one of three forms Transformation form: f (x) = a(x – h) 2 + k  Most useful for finding the vertex of a parabola  Vertex is at (h, k)  (Set inside parenthesis = 0 & solve, number outside)  If a is positive, the graph opens up.  If a is negative, graph opens down.  The y-intercept is at ah 2 + k  The x-intercepts are at

 Using Transformation Form Find the vertex of the function and state whether the graph opens upward or downward g(x) = -6(x – 2) 2 – 5 h(x) = -x 2 + 1 h = 2 and k = -5, so vertex is at (2, -5) Because a = -6, graph opens down There is no h, and k = 1 so vertex is at (0, 1) Because a = -1, graph opens down

 Polynomial form: f (x) = ax 2 + bx + c Yeah, we’ve seen this plenty already… Most useful for finding the y-intercept  y-intercept is at (0, c) If a is positive, the graph opens up. If a is negative, graph opens down. The vertex is at The x-intercepts are at

 Using Polynomial Form Determine the y-intercept and state whether the graph opens upward or downward g(x) = x 2 + 8x – 1 g(x) = 2x 2 – x + 5 The y-intercept is at (0, -1) Because a = 1, graph opens up The y-intercept is at (0, 5) Because a = 2, graph opens up

 x-intercept form: f (x) = a(x – s)(x – t) This is simply polynomial form factored out Most useful for finding the x-intercepts (duh)  x-intercepts are at (s, 0) and (t, 0) If a is positive, the graph opens up. If a is negative, graph opens down. The vertex is at The y-intercepts is at (0, ast)

 Using x-intercept Form Determine the x-intercepts and state whether the graph opens upward or downward h(x) = -2(x + 3)(x + 1) f(x) = -0.4(x + 2.1)(x – 0.7) The x-intercept are at (-3, 0) and (-1, 0) Because a = -2, graph opens down The x-intercepts are at (-2.1, 0) and (0.7, 0) Because a = -0.4, graph opens down

 Assignment Page 170 Problems 1-25, odd problems

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