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Agenda 1. WU (5 min) 2. Notes on graphing quadratics & properties of quadratics (40 min) WARM-UP

SWBAT… analyze the characteristics of the graphs of quadratic functions Wed, 6/3. Agenda 1. WU (5 min) 2. Notes on graphing quadratics & properties of quadratics (40 min) WARM-UP Place the path of a baseball (back of agenda) and Exponential Quiz in the blue folder

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Agenda 1. WU (5 min) 2. Notes on graphing quadratics & properties of quadratics (40 min) WARM-UP

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  1. SWBAT… analyze the characteristics of the graphs of quadratic functions Wed, 6/3 Agenda 1. WU (5 min) 2. Notes on graphing quadratics & properties of quadratics (40 min) WARM-UP • Place the path of a baseball (back of agenda) and Exponential Quiz in the blue folder 2. On two separate Cartesian Coordinate Systems: a.) Graph y = -x2 + 1 b.) Graph y = -x + 1 HW#1: Graphing Quadratic Functions

  2. 1.) How are the graphs of y = -x2 + 1 and y = -x + 1 different? Sample Answer: y = -x2 + 1 is a parabola that opens downward, while y = -x + 1 is a line that has a negative slope.

  3. Properties and Graphs of Quadratic Functions

  4. Standard form of a quadratic isy = ax2 + bx + c • a, b, and c are the coefficients • Example: • If y = 2x2 – 3x – 10, find a, b, and c • If 2x2 + 1 = -5x, find a, b, and c • When the power of an equation is 2, then the function is called a _________. quadratic

  5. y y = x2 . . . . . x Graphs of Quadratics parabola • The shape or graph of any quadratic equation is a ______________. • To graph a quadratic, set up a table and plot points Example: y = x2 x y -2 4 -1 1 0 0 1 1 2 4

  6. What are the steps to finding the solutions of a quadratic (Review) 1. Set the equation = 0 0 = ax2 + bx + c 2. Factor 3. Set each factor = 0 4. Solve for each variable 1)Algebraically (last week and next slide to review) 2)Graphically (today  in three slides)

  7. Directions: Find the zeros of the below function. f(x) = x2 – 8x + 12 0 = (x – 2)(x – 6) x – 2 = 0 or x – 6 = 0 x = 2 or x = 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

  8. Characteristics of Quadratic Functions • Parabolas are symmetric about a central line called the axis of symmetry(x = …) • 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 points where the parabola crosses the x-axis are the solutions or x-intercepts.

  9. Axis of symmetry y y-intercept vertex x-intercept x-intercept . . x 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)

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

  11. Ex: Graph y = -x2 + 1 (HW1 Prob #3) 1. Axis of symmetry: x = 0 2. Vertex: (0,1) y 3. y-intercept: (0, 1) 4. Solutions: x = 1 or x = -1 y = -x2 + 1 x y -2 -3 -1 0 0 1 1 0 2 -3 x

  12. Ex: Graph y = x2 – 4 (HW Prob #2) y 1. What is the axis of symmetry? y = x2- 4 x = 0 2. What is the vertex: (0, -4) x 3. What is the y-intercept: (0, -4) x y -2 0 -1 -3 0 -4 1 -3 2 0 4. What are the solutions: (x-intercepts) x = -2 or x = 2

  13. Finding the y-intercept Given y = ax2 + bx + c, what letter represents the y-intercept. Answer: c

  14. Given the below information, graph the quadratic function. • Axis of symmetry: x = 1 • Vertex: (1, 0) • Solutions: x = 1 (Double Root) • y-intercept: (0, 2) • Hint: The axis of symmetry splits the parabola in half

  15. y x x = 1 . (0, 2) . x = 1 (1, 0)

  16. SWBAT…analyze the characteristics of the graphs of quadratic functions 5/5 Agenda 1. WU (10 min) 2. Notes on graphing quadratics & properties of quadratics (30 min) WARM-UP: Graph y = x2 – 4 • What is the axis of symmetry? • What is the vertex? • What is the y-intercept? • What are the solutions? HW#1: Graphing Quadratic Functions

  17. SWBAT… analyze the characteristics and graphs of quadratic functions Wed, 5/11 1. WU (10 min) 2. Notes on axis of symmetry & vertex (20 min) 3. Work on hw1 (15 min) Warm-Up: Given the below information, graph the quadratic function. • Axis of symmetry: x = 1.5 • Vertex: (1.5, -6.25 ) • Solutions: x = -1 or x = 4 • y-intercept: (0, -5) HW#1: Graphing Quadratic Functions

  18. y x x = 1.5 . . x = -1 x = 4 . (0, -5) . (1.5, -6.25)

  19. Given the below information, graph the quadratic function. • Axis of symmetry: x = 1 • Vertex: (1, 0) • Solutions: x = 1 (Double Root) • y-intercept: (0, 2) • Hint: The axis of symmetry splits the parabola in half

  20. y x x = 1 . (0, 2) . x = 1 (1, 0)

  21. Calculating the Axis of Symmetry Algebraically Ex: Find the axis of symmetry of y = x2 – 4x + 7 a = 1 b = -4 c = 7

  22. Calculating the Vertex Algebraically Ex1: Find the vertex of y = x2 – 4x + 7 a = 1, b = -4, c = 7 y = x2 – 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 ( , )

  23. Ex3: (HW1 Prob #11) Find the vertex: y = 5x2 + 30x – 4 a = 5, b = 30 x = -b = -30 = -30 = -3 2a 2(5) 10 y = 5x2 + 30x – 4 y = 5(-3)2 + 30(-3) – 4 = -49 The vertex is at (-3, -49)

  24. Ex4 Vertex formula: Example: Find the vertex of y = 4x2 + 20x + 5 a = 4, b = 20, c = 5 y = 4x2 + 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 ( , )

  25. Ex5 Find the vertex: y = x2 + 4x + 7 a = 1, b = 4 x = -b = -4 = -4 = -2 2a 2(1) 2 y = x2 + 4x + 7 y = (-2)2 + 4(-2) + 7 = 3 The vertex is at (-2,3)

  26. Warm-Up: • Find the vertex: y = 2(x – 1)2+ 7 2(x – 1)(x – 1) + 7 2(x2 – 2x + 1) + 7 2x2 – 4x + 2 + 7 2x2 – 4x + 9 a = 2, b = -4, c = 9 y = 7 • Answer: (1, 7)

  27. Graphing Quadratic Functions • For your given quadratic find the following algebraically (show all work!): • Find the axis of symmetry • The vertex • Find the solutions • Find the y-intercept • After you find the above, graph the quadratic on graph paper

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