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-UP1. Write the hw in your planners

2. Review tests

3. Path of a baseball (back of agenda)

HW#1: Two Problems on graph paper

2nd Period Factoring Test Results

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Mean = 80% Range = 27% - 113%

Mean = 85% Range = 60% - 100%

4th Period Factoring Test Results

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Mean = 81% Range = 40% - 107%

6th Period Factoring Test Results

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All Infinity Algebra Classes Factoring Test Results

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Mean = 82% Range = 27% - 113%

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 Wednesday, 6/2 Thursday, 6/3 Friday, 6/4

NO CLASSES MEMORIAL DAY

Path of a baseball

Graphing & Properties of Quadratic Functions

HW#1

Graphing & Properties of Quadratic Functions

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 Review for Final

½ Day: A

Review for Final

FINAL EXAM!!!!

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 = ax2 + 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

Example: y = x2 x y

-2 4

-1 1

0 0

1 1

2 4

. .

..

.x

y

y = x2

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)

Axis of symmetry

.x-intercept x-intercept

.

vertexy-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)

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 quadratic1. Set y or f(x) equal to zero: 0 = ax2 + bx + c

2. Factor

3. Set each factor = 0

4. Solve for each variable1)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

y-intercept c

a

bx

2

Axis of symmetry examples

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

Example: y = x2 – 4

x

y

y = x2- 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

HOMEWORK #1 – On Graph Paper

1.) For y = -x2 + 11. Graph using a table of values

2. The axis of symmetry

3. The vertex

4. The y-intercept

5. The solutions (x-intercepts)

2.) How are the graphs of y = -x2 + 1 and y = -x + 1 different?

Example: y = -x2 + 1

x

y

y = -x2 + 1

2. Vertex:

4. x-intercepts:

3. y-intercept:

1. Axis of symmetry:

x y-2 -3 -1 0 0 1 1 0 2 -3

x = 0

(0,1)

(0,1)

x = 1 or x = -1

Vertex formulax = -b

2a

Steps to solve for the vertex:Step 1: Solve for x using x = -b/2aStep 2: Substitute the x-value in the original function to find the y-valueStep 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) + 7y = 2(x2 – 2x + 1) + 7y = 2x2 – 4x + 2 + 7y = 2x2 – 4x + 9a = 2, b = -4x = -(-4)/(2(2)) = 1y = 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)