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 1. 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 = -x 2 + 1 b.) Graph y = -x + 1 HW#1: Graphing Quadratic Functions

2. 1.) How are the graphs of y = -x 2 + 1 and y = -x + 1 different? Sample Answer: y = -x 2 + 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 is y = ax 2 + bx + c a, b, and c are the coefficients Example: If y = 2x 2 – 3x – 10, find a, b, and c If 2x 2 + 1 = -5x, find a, b, and c When the power of an equation is 2, then the function is called a _________. quadratic

5. Graphs of Quadratics The shape or graph of any quadratic equation is a ______________. 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 parabola

6. What are the steps to finding the solutions of a quadratic (Review) 1. Set the equation = 0 0 = ax 2 + 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) = x 2 – 8x + 12 0 = (x – 2)(x – 6) 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

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. 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)

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

11. Ex: Graph y = -x 2 + 1 (HW1 Prob #3) 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

12. Ex: Graph y = x 2 – 4 (HW Prob #2) 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

13. Finding the y-intercept Given y = ax 2 + bx + c, what letter represents the y-intercept. Answer: c

14. 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) 5. Hint: The axis of symmetry splits the parabola in half

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

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 = 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? 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. 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) HW#1: Graphing Quadratic Functions

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

19. 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) 5. Hint: The axis of symmetry splits the parabola in half

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

21. Calculating the Axis of Symmetry Algebraically Ex: Find the axis of symmetry of y = x 2 – 4x + 7 a = 1 b = -4 c = 7

22. 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 (, )

23. 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)

24. 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

25. 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)

26. Warm-Up: 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)

27. Graphing Quadratic Functions For your given quadratic find the following algebraically (show all work!): 1. Find the axis of symmetry 2. The vertex 3. Find the solutions 4. Find the y-intercept 5. After you find the above, graph the quadratic on graph paper