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Monday, 4/30Tuesday, 5/1Wednesday, 5/2Thursday, 5/3Friday, 5/4 No classes Review for tomorrow’s test TEST!Quadratic graphs Quadratic Graphs Monday, 5/7Tuesday, 5/8Wednesday, 5/9Thursday, 5/10Friday, 5/11 Quiz on quadratic graphs Quadratic Graphs Transformations of quadratic graphs Quadratic Formula Monday, 5/14Tuesday, 5/15Wednesday, 5/16Thursday, 5/17Friday, 5/18 Quadratic Formula Review for tomorrow’s test Test on quadratic graphs! Review for Exit Exam Monday, 5/22Tuesday, 5/23Wednesday, 5/24Thursday, 5/25Friday, 5/25 Review for Exit Exam EXIT EXAM!!1 (3PM – 5PM) Celebration!
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Monday, 5/14Tuesday, 5/15Wednesday, 5/16Thursday, 5/17Friday, 5/18 Review for tomorrow’s test Factoring & quadratic equations TEST Quadratic graphs Quadratic Graphs Monday, 5/21Tuesday, 5/22Wednesday, 5/23Thursday, 5/24Friday, 5/25 Quadratic graphs Transformations of quadratic graphs Quadratic Formula ½ Day: Junior Achievement Monday, 5/28Tuesday, 5/29Wednesday, 5/30Thursday, 5/31Friday, 6/1 No classes: Memorial Day Quadratic Formula Review for tomorrow’s test EXPLORE testing Test on quadratic functions! Monday, 6/4Tuesday, 6/5Wednesday, 6/6Thursday, 6/7Friday, 6/8 Review for final FINAL EXAM Part I FINAL EXAM Part II Monday, 6/11Tuesday, 6/12Wednesday, 6/13Thursday, 6/14Friday, 6/15 Assembly/ talent show/ Locker clean out ReflectionNo ClassesLast Day of School!!! VACATION! ELLADA!
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Graphs of Quadratics Let’s start by graphing the parent quadratic function y = x 2
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Graphs of Quadratics 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 4..... x y y = x 2
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Standard form of a quadratic y = ax 2 + bx + c a, b, and c are the coefficients Example: If y = 2x 2 – 3x + 10, find a, b, and c a = 2 b = -3 c = 10
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Characteristics of Quadratic Functions When the power of an equation is 2, then the function is called a quadratic function 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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In general equations have roots, Functions haves zeros, and Graphs of functions have x-intercepts
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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) maximum
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Axis of symmetry examples http://www.mathwarehouse.com/geometry/ parabola/axis-of-symmetry.php
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Given the below information, graph the quadratic function. 1. Axis of symmetry: x = 1.5 2. Vertex: (1.5, -6.25 ) 3. Solutions: x = -1 or x = 4 4. y-intercept: (0, -5)
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x y... (0, -5) x = 4 x = -1 x = 1.5. (1.5, -6.25)
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Given the below information, graph the quadratic function. 1. Axis of symmetry: x = 1 2. Vertex: (1, 0) 3. Solutions: x = 1 (Double Root) 4. y-intercept: (0, 2) Hint: The axis of symmetry splits the parabola in half
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x y. (1, 0) x = 1. (0, 2)
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Graph y = x 2 – 4 1. What is the axis of symmetry? 2. What is the vertex? 3. What is the y-intercept? 4. What are the solutions? 5. What is the domain? 6. What is the range?
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Ex: Graph y = x 2 – 4 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 (0, -4) x = 0 5. What is the domain? All real numbers 6. What is the range? y ≥ -4
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Finding the y-intercept Given y = ax 2 + bx + c, what letter represents the y-intercept. Answer: c
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Calculating the Axis of Symmetry Algebraically Ex: Find the axis of symmetry of y = x 2 – 4x + 7 a = 1 b = -4 c = 7
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Calculating the Vertex Algebraically Ex1: Find the vertex of y = x 2 – 4x + 7 a = 1, b = -4, c = 7 y = x 2 – 4x + 7 y = (2) 2 – 4(2) + 7 = 3 The vertex is at (2, 3) 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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Ex3: (HW1 Prob #11) 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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Vertex formula: Example: Find the vertex of y = 4x 2 + 20x + 5 a = 4, b = 20, c = 5 y = 4x 2 + 20x + 5 y = 4(-2.5) 2 + 20(-2.5) + 5 = -20 The vertex is at (-2.5,-20) 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 (, ) Ex4 (HW1 Prob #9)
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Ex5 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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Find the vertex: y = 2(x – 1) 2 + 7 2(x – 1)(x – 1) + 7 2(x 2 – 2x + 1) + 7 2x 2 – 4x + 2 + 7 2x 2 – 4x + 9 a = 2, b = -4, c = 9 y = 7 Answer: (1, 7) (HW1 Prob #12)
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SWBAT… graph quadratic functions. Mon, 5/21 Agenda 1. WU (15 min) 2. Graphs of quadratic functions - posters (30 min) Warm-Up: 1. Take out HW#1: Any questions? 2. Review the weekly agenda HW#2: Quadratic functions (both sides)
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HW1, Problem #4 Axis of symmetry: x = -2 Vertex: (-2, -1) y-intercept: (0, 3) Solutions: x = -3 or x = -1 Domain: All real numbers Range: y ≥ -1
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Graph y = -x 2 + 1 (HW1 Prob #2) x y y = -x 2 + 1 2. Vertex: (0,1) 4. Solutions: x = 1 or x = -1 3. y-intercept: (0, 1) 1. Axis of symmetry: x = 0 x y -2 -3 -1 0 0 1 1 0 2 -3 5. What is the domain? 6. What is the range? All real numbers y ≤ 1
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Vertex formula: Example: Find the vertex of y = 4x 2 + 20x + 5 a = 4, b = 20, c = 5 y = 4x 2 + 20x + 5 y = 4(-2.5) 2 + 20(-2.5) + 5 = -20 The vertex is at (-2.5,-20) 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 (, ) Ex4 (HW1 Prob #9)
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Ex3: (HW1 Prob #11) 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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Find the vertex: y = 2(x – 1) 2 + 7 2(x – 1)(x – 1) + 7 2(x 2 – 2x + 1) + 7 2x 2 – 4x + 2 + 7 2x 2 – 4x + 9 a = 2, b = -4, c = 9 y = 7 Answer: (1, 7) (HW1 Prob #12)
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Graphing Quadratic Functions For your given quadratic find the following algebraically (show all work on poster!): 1. Find the axis of symmetry 2. The vertex (ordered pair) 3. Find the solutions 4. Find the y-intercept (ordered pair) 5. After you find the above, graph the quadratic on graph paper 6. Find the domain 7. Find the range (need the vertex!)
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3 Types of Solutions 1. Two real roots 1. Parabolas crosses two different points on the x-axis 2. Double root 1. Parabola crosses the same point on the x-axis 3. No real roots 1. Parabola does not cross the x-axis
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Exit Slip: Complete on graph paper: Given y = x 2 + 6x + 8 find algebraically: 1. The axis of symmetry 2. The vertex (as an ordered pair) 3. The solutions (x-intercepts) 4. The y-intercept (as an ordered pair) 5. After you find the above, graph the quadratic 6. Domain 7. Range
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