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Warm-Up Find the vertex, the roots or the y- intercept of the following forms: 1. f(x) = (x-4) 2 -1 2. f(x) = -2(x-3)(x+4) 3. f(x) = x 2 -2x -15 Answers: 1. Vertex (4,-1) 2. Roots are x=3, x= -4 3.Y-int: (0,-15) vertex: (1, -16) Classwork: Pg. 309 (3-36 every 3 rd one)

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Lesson 3.1A Quadratic Functions– The Three Forms 1. Standard or General Form y = ax 2 + bx + c This form tells me the __________________. That is where the graph ________________. The y-intercept is _____________________. y-intercept crosses the y-axis (0,c)

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2. Factored Form y = a( x – r 1 ) (x-r 2 ) This form tells me the __________________. That is where the graph ________________. To find the roots, zeros or x-intercepts _______________. roots, zeros, or x-intercepts crosses the x-axis Set ( x – r 1 ) (x-r 2 ) = 0

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3. Vertex Form y = a( x –h) 2 +k This form tells me the __________________. That is where the graph ________________. The vertex is _____________________. vertex Minimum or maximum of the parabola (h,k)

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General Form Factored Form Vertex Form How to convert from different forms? Given:

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Given:

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Ex. 1: y = x 2 -6x + 5 What form? Change from general to ____________ by using either _______________ or ___________________. General Form Factored Form Factoring Quadratic Formula X 2 – 6x + 5 (x-5)(x-1) = 0 X = 5, x = 1

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Ex. 2: y = x 2 -6x + 5 Change from general form to __________ by using _________________________. Vertex form Completing the square y = x 2 -6x + 5 Complete the square Y=(x 2 -6x + _____) +5 - _____ Y=(x-3) 2 -4 Now Graph: Y-intercept:_______ Vertex :_______ X-intercepts: ______, ______ (0,5) (3,-4) (5,0) (1,0) 99

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Ex. 3: Using f(x) = -3x 2 +6x - 13 find the vertex when given the standard form? Use the formula to find x : Substitute in x to find y. In standard form a = -3, b = 6, c= -13 x = 1, then y = -3(1) 2 +6(1) -13 = -10 Vertex = (1, -10)

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You try: y = (x+4) 2 -13 Answers: GF: x 2 +8x+3 FF: y = (x+.39) (x+7.61) HINT: If you cannot factor it, you must use the Quadratic Formula!!

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Ex. 4: Minimum and Maximum Quadratic Functions Consider the quadratic function: f(x) = -3x 2 + 6x – 13 a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the max. or min. value and determine where it occurs c. Identify the function’s domain and range.

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Ex. 4 Continued f(x) = -3x 2 + 6x – 13 Begin by identifying a, b, and c. a. Because a 0 then the function would have a min. b. The max. occurs at x = -b/2a = - 6 /2(-3) = -6/-6 = 1. The maximum value occurs at x = 1 and the maximum value of f(x) = -3x 2 + 6x – 13 f(1) = -3*1 2 + 6*1 – 13 = -3 + 6 – 13=-10 Plug in one for x into original function. We see that the max is -10 at x = 1. c. Domain is (-∞,∞) Range (-∞,-10].

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YOU TRY!! Repeat parts a through c using the function: f(x) = 4x 2 – 16x + 1000. Answer: a. Min b. Min is 984 at x = 2 c. Domain is (-∞,∞) Range is [984,∞)

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Summary: Describe how to find a parabola’s vertex if its equation is expressed in standard form.

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