1. Graphs of Quadratics Let’s start by graphing the parent quadratic function y = x 2

2. 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

3. 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

4. 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

5. In general equations have roots, Functions haves zeros, and Graphs of functions have x-intercepts

6. 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

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

8. 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)

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

10. 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

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

12. 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?

13. 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

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

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

16. 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 (, )

17. 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)

18. 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)

19. 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)

20. 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)

21. 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)

22. 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

23. 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

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 (HW1 Prob #9)

25. 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)

26. 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)

27. 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!)

28. 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