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Monday, 5/10Tuesday, 5/11Wednesday, 5/12Thursday, 5/13Friday, 5/14 Graphing & Properties of Quadratic Functions HW#1 Graphing & Properties of Quadratic Functions HW#1 Solving Quadratic Equations by Graphing Path of a Baseball HW#2 ½ Day: B Activity on big graph paper: Graphing Quadratics HW#3 (quiz) TI-84 Graphing Calculator Investigation Activity: Transformations of Quadratics HW#4 Monday, 5/17Tuesday, 5/18Wednesday, 5/19Thursday, 5/20Friday, 5/21 Solving Quadratic Equations by Using Completing the Square HW#5 Solving Quadratic Equations by Using Completing the Square ½ Day: A Solving Quadratic Equations by Using the Quadratic Formula HW#6 Quiz: Completing the Square & Quadratic Formula Additional practice – quadratic formula & completing the square HW#7 Monday, 5/24Tuesday, 5/25Wednesday, 5/26Thursday, 5/27Friday, 5/28 Review for test Test: Factoring & Quadratic Functions ???Rocket Project???

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Graphs of Quadratics

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Terms of a quadratic y = ax 2 + bx + c Every quadratic has terms: Quadratic term: ax 2 Linear term: bx Constant term: c When the power of an equation is 2, then the function is called a quadratic a, b, and c are the coefficients Standard form of a quadratic

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Graphs of Quadratics The graph of any quadratic equation is a parabola To graph a quadratic, set up a table and plot points Example: y = x 2 x y -2 4 -1 1 0 0 1 1 2 2..... x y y = x 2

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Finding the solutions of a quadratic 2. Find the values of x that make the equation equal to 0 1)Algebraically (last week and next slide to review) 2)Graphically (today next slide) 1. Set y of f(x) equal to zero: 0 = ax 2 + bx + c In general equations have roots, Functions haves zeros, and Graphs of functions have x-intercepts

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Directions: Find the zeros. Ex: f(x) = x 2 – 8x + 12 Factor and set y or f(x) = 0 (x – 2)(x – 6) = 0 x – 2 = 0 or x – 6 = 0 x = 2 orx = 6 Factors of 12 Sum of Factors, -8 1, 12 13 2, 6 8 3, 4 7 -1, -12 -13 -2, -6 -8 -3, -4 -7

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Characteristics of Quadratic Functions The shape of a graph of a quadratic function is called a parabola. Parabolas are symmetric about a central line called the axis of symmetry. The axis of symmetry intersects a parabola at only one point, called the vertex. The lowest point on the graph is the minimum. The highest point on the graph is the maximum. The maximum or minimum is the vertex

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Axis of symmetry. x-intercept. vertex y-intercept x y Characteristics of Quadratic Functions To find the solutions graphically, look for the x-intercepts of the graph (Since these are the points where y = 0)

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Key Concept: Quadratic Functions Parent Functionf(x) = x 2 Standard From f(x) = ax 2 + bx + c Type of GraphParabola Axis of Symmetry y-interceptc

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Axis of symmetry examples http://www.mathwarehouse.com/geometry/ parabola/axis-of-symmetry.php

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Vertex formula x = -b 2a Steps to solve for the vertex: Step 1: Solve for x using x = -b/2a Step 2: Substitute the x-value in the original function to find the y-value Step 3: Write the vertex as an ordered pair (, )

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Example 1: HW Prob #11 Find the vertex: y = 4x 2 + 20x + 5 a = 4, b = 20 x = -b = -20 = -20 = -2.5 2a 2(4) 8 y = 4x 2 + 20x + 5 y = 4(-2.5) 2 + 20(-2.5) + 5 = -20 The vertex is at (-2.5,-20)

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Example 2 Find the vertex: y = x 2 – 4x + 7 a = 1, b = -4 x = -b = -(-4) = 4 = 2 2a 2(1) 2 y = x 2 – 4x + 7 y = (2) 2 – 4(2) + 7 = 3 The vertex is at (2,3)

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Example 3: HW Prob #14 Find the vertex: y = 5x 2 + 30x – 4 a = 5, b = 30 x = -b = -30 = -30 = -3 2a2(5) 10 y = 5x 2 + 30x – 4 y = 5(-3) 2 + 30(-3) – 4 = -49 The vertex is at (-3,-49)

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Example 4 Find the vertex: y = 2(x-1) 2 + 7 Answer: (1, 7)

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Example 5 Find the vertex: y = x 2 + 4x + 7 a = 1, b = 4 x = -b = -4 = -4 = -2 2a 2(1) 2 y = x 2 + 4x + 7 y = (-2) 2 + 4(-2) + 7 = 3 The vertex is at (-2,3)

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Example: y = x 2 – 4 (HW Prob #1) x y y = x 2 - 4 2. What is the vertex (, ) 4. What are the solutions: (x-intercepts) 3. What is the y-intercept: 1. What is the axis of symmetry? x y -2 0 -1 -3 0 -4 1 -3 2 0 (0, -4) x = -2 or x = 2 -4 x = 0

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Example: y = -x 2 + 1 (HW Prob #3) x y y = -x 2 + 1 2. Vertex: (0,1) 3. x-intercepts: x = 1 or x = -1 4. y-intercept: 1 1. Axis of symmetry: x = 0 x y -2 -3 -1 0 0 1 1 0 2 -3

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