Monday, 5/10 Tuesday, 5/11 Wednesday, 5/12 Thursday, 5/13 Friday, 5/14

Graphing & Properties of

Quadratic Functions

HW#1

Graphing & Properties of Quadratic Functions

HW#1

Solving Quadratic Equations by Graphing

Path of a Baseball

HW#2

½ Day: B

Activity on big graph paper: Graphing Quadratics

HW#3 (quiz)

TI-84 Graphing Calculator Investigation Activity: Transformations of Quadratics

HW#4

Monday, 5/17 Tuesday, 5/18 Wednesday, 5/19 Thursday, 5/20 Friday, 5/21

Solving Quadratic Equations by Using Completing the Square

HW#5

Solving Quadratic Equations by Using Completing the Square ½ Day: A

Solving Quadratic Equations by Using the

Quadratic Formula

HW#6

Quiz: Completing the Square & Quadratic Formula

Additional practice – quadratic formula & completing the square

HW#7

Monday, 5/24 Tuesday, 5/25 Wednesday, 5/26 Thursday, 5/27 Friday, 5/28

Review for test Test: Factoring & Quadratic

Functions

???Rocket Project???

Graphs of Quadratics

Terms of a quadratic y = ax2 + bx + c

Every quadratic has terms: Quadratic term: ax2

Linear term: bx Constant term: c

When the power of an equation is 2, then the function is called a quadratic

a, b, and c are the coefficients

Standard form of 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 2

. .

..

.x

y

y = x2

Finding the solutions of a quadratic

2. Find the values of x that make the equation equal to 01)Algebraically (last week and next slide to review)

2)Graphically (today next slide)

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

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

Factor and set y or f(x) = 0

(x – 2)(x – 6) = 0

x – 2 = 0 or x – 6 = 0

x = 2 or x = 6Factors of 12

Sum of Factors, -8

1, 12 13

2, 6 8

3, 4 7

-1, -12 -13

-2, -6 -8

-3, -4 -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

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)

Key Concept: Quadratic Functions

Parent Function f(x) = x2

Standard From 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

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: HW Prob #11

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 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)

Example 3: HW Prob #14

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)

Example 4

Find the vertex: y = 2(x-1)2 + 7

Answer: (1, 7)

Example 5

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: y = x2 – 4 (HW Prob #1)

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

-4

x = 0

Example: y = -x2 + 1 (HW Prob #3)

x

y

y = -x2 + 1

2. Vertex: (0,1)3. x-intercepts: x = 1 or x = -1

4. y-intercept: 1

1. Axis of symmetry: x = 0

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