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Family of Quadratic Functions Lesson 5.5a
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General Form Quadratic functions have the standard form y = ax 2 + bx + c a, b, and c are constants a ≠ 0 (why?) Quadratic functions graph as a parabola
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Zeros of the Quadratic Zeros are where the function crosses the x- axis Where y = 0 Consider possible numbers of zeros None (or two complex) One Two
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Axis of Symmetry Parabolas are symmetric about a vertical axis For y = ax 2 + bx + c the axis of symmetry is at Given y = 3x 2 + 8x What is the axis of symmetry?
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Vertex of the Parabola The vertex is the “point” of the parabola The minimum value Can also be a maximum What is the x-value of the vertex? How can we find the y-value?
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Vertex of the Parabola Given f(x) = x 2 + 2x – 8 What is the x-value of the vertex? What is the y-value of the vertex? The vertex is at (-1, -9)
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Vertex of the Parabola Given f(x) = x 2 + 2x – 8 Graph shows vertex at (-1, -9) Note calculator’s ability to find vertex (minimum or maximum)
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Shifting and Stretching Start with f(x) = x 2 Determine the results of transformations ___ f(x + a) = x 2 + 2ax + a 2 ___ f(x) + a = x 2 + a ___ a * f(x) = ax 2 ___ f(a*x) = a 2 x 2 a) horizontal shift b) vertical stretch or squeeze c) horizontal stretch or squeeze d) vertical shift e) none of these
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Other Quadratic Forms Standard form y = ax 2 + bx + c Vertex form y = a (x – h) 2 + k Then (h,k) is the vertex Given f(x) = x 2 + 2x – 8 Change to vertex form Hint, use completing the square Experiment with Quadratic Function Spreadsheet
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Vertex Form Changing to vertex form Add something in to make a perfect square trinomial Subtract the same amount to keep it even. Now create a binomial squared This gives us the ordered pair (h,k)
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Assignment Lesson 5.5a Page 231 Exercises 1 – 25 odd
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