1. HW#1: Two Problems on graph paper
SWBAT… analyze the characteristics of the graphs of quadratic functions /2/10
Agenda
1. WU (15 min)
2. Notes on graphing quadratics & properties of quadratics (30 min)
WARM-UP
Write the hw in your planners
Review tests
Path of a baseball (back of agenda)
HW#1: Two Problems on graph paper

2. Mean = 80%
Range = 27% - 113%

3. Mean = 85%
Range = 60% - 100%

4. Mean = 81%
Range = 40% - 107%

5. Mean = 82%
Range = 27% - 113%

6. To get us warmed up and ready for quadratics…
Complete the path of a baseball on the back of this week’s agenda

7. Solving Quadratic Equations by Using the Quadratic Formula
Monday, 5/31
Tuesday, 6/1
Wednesday, 6/2
Thursday, 6/3
Friday, 6/4
NO CLASSES
MEMORIAL DAY
Path of a baseball
Graphing & Properties of Quadratic Functions
HW#1
HW#2
Activity on big graph paper: Graphing Quadratics
HW#3 (quiz)
TI-84 Graphing Calculator Investigation Activity: Transformations of Quadratics
HW#4
Monday, 6/7
Tuesday, 6/8
Wednesday, 6/9
Thursday, 6/10
Friday, 6/11
Solving Quadratic Equations by Using the Quadratic Formula
HW#5
Review for Final
½ Day: A
FINAL EXAM!!!!
Monday, 6/14
Tuesday, 6/15
Wednesday, 6/16
Thursday, 6/17
Friday, 6/18
TBA
Last Day of Class!

8. Properties of Quadratic Functions
Agenda:
Standard form of a quadratic (1 slide)
Graphing quadratics (1 slide)
Finding solutions to quadratics (1 slide)
Characteristics of quadratic functions (3 slides)
Quadratic graphs examples (1 slide)
HW Problem

9. Standard form of a quadratic
y = ax2 + bx + c
When the power of an equation is 2, then the function is called a quadratic

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

11. HOMEWORK #1 – On Graph Paper (Warm Up – will be collected)
1.) Graph y = -x2 + 1 using a table of values (answer on the front side)
2.) How are the graphs of y = -x2 + 1 and y = -x + 1 different? (answer on the back side)

12. Characteristics of Quadratic Functions
Axis of symmetry .
x-intercept
vertex
y-intercept x y
To find the solutions graphically, look for the x-intercepts of the graph
(Since these are the points where y = 0)

13. 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 (ordered pair).
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

14. Finding the solutions of a quadratic
1. Set y or f(x) equal to zero: 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 a few slides)
In general equations have roots,
Functions haves zeros, and
Graphs of functions have x-intercepts

15. Ex: f(x) = x2 – 8x + 12 0 = (x – 2)(x – 6) x – 2 = 0 or x – 6 = 0
Directions: Find the zeros.
Ex: f(x) = x2 – 8x + 12
0 = (x – 2)(x – 6)
x – 2 = 0 or x – 6 = 0
x = or x = 6
Factors of 12
Sum of Factors, -8 1, 2, 3,
-1,
-2,
-3,

16. Key Concept: Quadratic Functions
Parent Function f(x) = x2
Standard Form f(x) = ax2 + bx + c
Type of Graph Parabola
Axis of Symmetry
y-intercept c

17. Axis of symmetry examples

18. Example: y = x2 – 4 1. What is the axis of symmetry?
2. What is the vertex:
(0, -4) x
3. What is the y-intercept:
(0, -4)
x y
2 0
1. (0, -4)
2. x = 2 or x = -2
3. -4
4. x = 0
4. What are the solutions:
(x-intercepts)
x = -2 or x = 2

19. HOMEWORK #1 – On Graph Paper
1.) For y = -x2 + 1
Graph using a table of values
The axis of symmetry
The vertex
The y-intercept
The solutions (x-intercepts)
2.) How are the graphs of y = -x and y = -x + 1 different?

20. Example: y = -x2 + 1 1. Axis of symmetry: 2. Vertex: 3. y-intercept:
(0,1)
2. Vertex: y
3. y-intercept:
(0,1)
4. x-intercepts:
x = 1 or x = -1
y = -x2 + 1
x y x
1. (0, 1)
2. x = 1 or x = -1
3. 1
4. x = 0

21. Vertex formula x = -b 2a 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 ( , )

22. Example 1 Find the vertex: y = x2 – 4x + 7 x = -b = -(-4) = 4 = 2
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)

23. Example 2 Find the vertex: y = x2 + 4x + 7 x = -b = -4 = -4 = -2
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)

24. HOMEWORK Find the vertex: y = 2(x – 1)2 + 7 y = 2(x – 1)(x – 1) + 7
a = 2, b = -4
x = -(-4)/(2(2)) = 1
y = 2(1 – 1)2 + 7
y = 2(0)2 + 7
Answer: (1, 7)

25. Example 3: (HW1 Prob #9) Find the vertex: y = 4x2 + 20x + 5
a = 4, b = 20
x = -b = -20 = -20 = -2.5
2a 2(4) 8
y = 4x2 + 20x + 5
y = 4(-2.5)2 + 20(-2.5) + 5 = -20
The vertex is at (-2.5,-20)

26. Example 4: (HW1 Prob #12) Find the vertex: y = 5x2 + 30x – 4
a = 5, b = 30
x = -b = -30 = -30 = -3
2a 2(5)
y = 5x2 + 30x – 4
y = 5(-3)2 + 30(-3) – 4 = -49
The vertex is at (-3,-49)