1. SWBAT…analyze the characteristics of the graphs of quadratic functions Wed, 2/15 Agenda 1. WU (10 min) 2. Characteristics of quadratic equations (35 min) Warm-Up: Given the function f(x) = 3x 2 – 18x + 15, for what values of x does f(x) = 0. HW#9: Quadratics

2. Quadratic Functions and their Graphs

3. Standard form of a quadratic y = ax 2 + bx + c a, b, and c are the coefficients Example: y = 2x 2 – 3x + 10 a = 2 b = -3 c = 10 When the power of an equation is 2, then the function is called a quadratic function

4. Quadratic Functions and their Graphs 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 4..... x y y = x 2

5. Finding the solutions of a quadratic (Review) 1. Set the equation = 0 2. Set y or f(x) equal to zero: 0 = ax 2 + bx + c 3. Factor 4. Set each factor = 0 5. Solve for each variable 1)Algebraically (last week and next slide to review) 2)Graphically (today  in three slides) In general equations have roots, Functions haves zeros, and Graphs of functions have x-intercepts

6. 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

7. 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

8. 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

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

10. 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) (HW9 Prob #8)

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

12. WU: From the HW graph: x y 2. What is the vertex: 4. What are the solutions: (x-intercepts) 5. What is the domain? 6. What is the range? 3. What is the y-intercept: 1. What is the axis of symmetry?

13. WU: Graph y = x 2 – 4 x y 2. What is the vertex: 4. What are the solutions: (x-intercepts) 5. What is the domain? 6. What is the range? 3. What is the y-intercept: 1. What is the axis of symmetry?

14. Ex: Graph y = x 2 – 4 x y y = x 2 - 4 2. What is the vertex: 4. What are the solutions: (x-intercepts) 5. What is the domain? 6. What is the range? 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

15. Ex: Graph y = -x 2 + 1 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. Domain: All real numbers 6. Range: y ≤ 1

16. 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

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

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

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

20. Calculating the Vertex (x, y) 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 (, )

21. Ex: Graph y = -x 2 – 3 x y 2. What is the vertex: 4. What are the solutions: (x-intercepts) 5. What is the domain? 6. What is the range? 3. What is the y-intercept: 1. What is the axis of symmetry?

22. Ex2: (HW9 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)

23. 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 (, ) Ex3: (HW5 Prob #9)

24. Ex4: 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)

25. Find the vertex: y = 2(x – 1) 2 + 7 y = 2(x – 1)(x – 1) + 7 y = 2(x 2 – 2x + 1) + 7 y = 2x 2 – 4x + 2 + 7 y = 2x 2 – 4x + 9 a = 2, b = -4, c = 9 y = 7 Answer: (1, 7) (HW5 Prob #12)

26. Given y = x 2 + 6x + 8, find the following algebraically 1. Axis of symmetry 2. Vertex (as an ordered pair) 3. Solutions (x-intercepts) 4. y-intercept (as an ordered pair) 5. After finding the above, graph the function 6. Domain 7. Range