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ParabolasParabolas by Dr. Carol A. Marinas

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Transformation Shifts Tell me the following information about: f(x) = (x – 4) 2 – 3 What shape is the graph? What direction is it going? What is the vertex? Is it a high point or low point? What is the y-intercept?

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Transformation Shifts Tell me the following information about: f(x) = (x – 4) 2 – 3 What shape is the graph? parabola What direction is it going? up What is the vertex? (4, –3) Is it a high point or low point? Low point What is the y-intercept? (0, 13)

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y =(x – 4) 2 – 3

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Standard Form y = (x – h) 2 + k Vertex is (h, k) Line of Symmetry is x = h

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Standard to General Form y =(x – 4) 2 – 3 y = (x 2 – 8x + 16) – 3 y = x 2 – 8x + 13

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General to Standard Form y = x 2 – 8x + 13 y – 13 = x 2 – 8x y – 13 + 16 = x 2 – 8x + 16 y + 3 = (x – 4) 2 y = (x – 4) 2 – 3 Vertex is (4, – 3) Get ‘a’ equal to 1 by multiplying or dividing the equation. (done) Move constant to left side Complete the square Solve for y

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General Form y = ax 2 + bx + c Vertex is ( –b/2a, f(–b/2a) ) y-intercept is (0, c) Line of Symmetry is x = –b/2a Example: y = x 2 – 8x + 13 Vertex is ( –(– 8)/2(1), f (8/2)) or (4, f(4)) or (4, –3) y-intercept is (0, 13) Line of Symmetry is x = 4

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x-intercepts For x-intercepts, the y value is 0. y = ax 2 + bx + c becomes 0 = ax 2 + bx + c which is a quadratic equation that is solved by factoring or the quadratic formula.

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x-intercepts using Quadratic Formula 0 = ax 2 + bx + c x = b 2 – 4ac is the discriminant and is used to tell us how many x-intercepts exist.

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Discriminant if less than 0, no x-intercepts b 2 – 4ac if it is 0, 1 x-intercept if greater than 0, 2 x-intercepts Example: y = x 2 – 8x + 13 The discriminant is (–8) 2 – 4(1)(13) or 64 – 52 or 12 Since 12 > 0, there are 2 x-intercepts.

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Finding the x-intercepts Ex: y = x 2 – 8x + 13 The discriminant is 12. To actually find the x-intercepts, let’s continue using the Quadratic Formula. x = = x = 8 ± √12 = 8 ± 2√3 = 4 ± 2 2 The x-intercepts are (4 –, 0 ) and (4 +, 0)

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Final Graph of y = (x – 4) 2 – 3 or y = x 2 – 8x + 13

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ReviewReview Standard Form y = (x – h) 2 + k General Form y = ax 2 + bx + c Know how to find the following: * Vertex * y-intercept(s) * High/Low Point * x-intercept(s) * Axis of Symmetry * Graph the parabola

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