SWBAT… analyze the characteristics of the graphs of quadratic functions
6/2/10
Agenda
1. WU (15 min)
2. Notes on graphing quadratics & properties of quadratics (30 min)
WARM-UP
2.
Write the hw in your planners
Review tests
3.
Path of a baseball (back of agenda)
1.
HW#1: Two Problems on graph paper
Number of Students
2nd Period Factoring Test Results
7
6
6
5
4
4
4
3
3
2
2
1
0
A
B
C
Test Grade
Mean = 80%
 Range = 27% - 113%

D
F
Number of Students
4th Period Factoring Test Results
6
5
5
4
4
3
2
2
1
1
0
0
A
B
C
Test Grade
Mean = 85%
 Range = 60% - 100%

D
F
Number of Students
6th Period Factoring Test Results
8
7
6
5
4
3
2
1
0
7
6
1
0
A
B
C
Test Grade
Mean = 81%
 Range = 40% - 107%

6
D
F
Number of Students
All Infinity Algebra Classes Factoring Test Results
20
18
15
12
9
10
9
3
5
0
A
B
C
Test Grade
Mean = 82%
 Range = 27% - 113%

D
F
To get us warmed up and ready for
quadratics…

Complete the path of a baseball on the
back of this week’s agenda
Monday, 5/31
Tuesday, 6/1
Path of a baseball
NO CLASSES
MEMORIAL DAY
Wednesday, 6/2
Graphing & Properties
of Quadratic Functions
Graphing &
Properties of
Quadratic Functions
Thursday, 6/3
Activity on big graph
paper: Graphing
Quadratics
Friday, 6/4
TI-84 Graphing
Calculator
Investigation Activity:
Transformations of
Quadratics
HW#1
HW#2
HW#3 (quiz)
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
Review for Final
Review for Final
Review for Final
FINAL
EXAM!!!!
½ Day: A
HW#5
Monday, 6/14
Tuesday, 6/15
Wednesday, 6/16
Thursday, 6/17
Friday, 6/18
TBA
TBA
TBA
NO CLASSES
Last Day of Class!
Properties of Quadratic
Functions
Agenda:
1. Standard form of a quadratic (1 slide)
2. Graphing quadratics (1 slide)
3. Finding solutions to quadratics (1 slide)
4. Characteristics of quadratic functions (3 slides)
5. Quadratic graphs examples (1 slide)
6. HW Problem
Standard form of a quadratic
y=

2
ax +
bx + c
When the power of an equation is 2, then the
function is called a quadratic
Graphs of Quadratics


The graph of any quadratic equation is a parabola
To graph a quadratic, set up a table and plot points
y
Example: y = x2
x y
y = x2
-2 4
. .
-1 1
...
x
0 0
1 1
2 4
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)
Characteristics of Quadratic Functions
Axis of symmetry
y
vertex
y-intercept
x-intercept
x-intercept
.
To find the solutions graphically, look
for the x-intercepts of the graph
(Since these are the points where y = 0)
.
x
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
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
Directions: Find the zeros.
Ex: 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
Key Concept: Quadratic Functions
Parent Function
f(x) = x2
Standard Form
f(x) = ax2 + bx + c
Type of Graph
Parabola
Axis of Symmetry
b
x
2a
y-intercept
c
Axis of symmetry examples

http://www.mathwarehouse.com/geometry/
parabola/axis-of-symmetry.php
2
x –
Example: y =
y
y=
x2-
1. What is the axis of
symmetry? x = 0
4
x
x
y
-2
0
-1 -3
0
-4
1
-3
2
0
4
2. What is the vertex: (0, -4)
3. What is the y-intercept: (0, -4)
4. What are the solutions:
(x-intercepts) x = -2 or x = 2
HOMEWORK #1 – On Graph Paper
1.) For y =
1.
2.
3.
4.
5.
2
-x +
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 = -x2 + 1
and y = -x + 1 different?
Example: y =
2
-x +
1
1. Axis of symmetry: x = 0
y
2. Vertex: (0,1)
3. y-intercept: (0,1)
y = -x2 + 1
4. x-intercepts: x = 1 or x = -1
x
x
-2
y
-3
-1
0
0
1
1
0
2
-3
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 ( , )
Example 1
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)
Example 2
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)
HOMEWORK
 Find
the vertex: y = 2(x – 1)2 + 7
y = 2(x – 1)(x – 1) + 7
y = 2(x2 – 2x + 1) + 7
y = 2x2 – 4x + 2 + 7
y = 2x2 – 4x + 9
a = 2, b = -4
x = -(-4)/(2(2)) = 1
y = 2(1 – 1)2 + 7
y = 2(0)2 + 7

Answer: (1, 7)
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)
Example 4: (HW1 Prob #12)
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)