SWBAT… analyze the characteristics of the graphs of quadratic functions Wed, 6/3

Agenda

1. WU (5 min)

2. Notes on graphing quadratics & properties of quadratics (40 min)

WARM-UP1. Place the path of a baseball (back of agenda) and

Exponential Quiz in the blue folder

2. On two separate Cartesian Coordinate Systems:

a.) Graph y = -x2 + 1

b.) Graph y = -x + 1

HW#1: Graphing Quadratic Functions

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

Sample Answer: y = -x2 + 1 is a parabola that opens downward, while y = -x + 1 is a line that has a negative slope.

Properties and Graphs of Quadratic

Functions

Standard form of a quadratic isy = ax2 + bx + c

a, b, and c are the coefficients Example:

If y = 2x2 – 3x – 10, find a, b, and c If 2x2 + 1 = -5x, find a, b, and c

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

Graphs of Quadratics The shape or graph of any quadratic equation is a

______________. 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

parabola

What are the steps to finding the solutions of a quadratic (Review)

1. Set the equation = 0 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 three slides)

Directions: Find the zeros of the below function.

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

Characteristics of Quadratic Functions Parabolas are symmetric about a central line

called the axis of symmetry (x = …) 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 points where the parabola crosses the x-

axis are the solutions or x-intercepts.

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)

Axis of symmetry examples

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

Ex: Graph y = -x2 + 1 (HW1 Prob #3)

x

y

y = -x2 + 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

Ex: Graph y = x2 – 4 (HW Prob #2)

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

Finding the y-intercept

Given y = ax2 + bx + c, what letter represents the y-intercept.

Answer: c

Given the below information, graph the quadratic function.

1. Axis of symmetry: x = 12. Vertex: (1, 0)3. Solutions: x = 1 (Double Root)4. y-intercept: (0, 2)

5. Hint: The axis of symmetry splits the parabola in half

x

y

.(1, 0)x = 1

x = 1

.(0, 2)

SWBAT…analyze the characteristics of the graphs of quadratic functions 5/5

Agenda 1. WU (10 min)2. Notes on graphing quadratics & properties of quadratics (30 min)

WARM-UP:

Graph y = x2 – 41. What is the axis of symmetry?2. What is the vertex?3. What is the y-intercept?4. What are the solutions?

HW#1: Graphing Quadratic Functions

SWBAT… analyze the characteristics and graphs of quadratic functions Wed, 5/11

1. WU (10 min)2. Notes on axis of symmetry & vertex (20 min)3. Work on hw1 (15 min)

Warm-Up:Given the below information, graph the quadratic function.1. Axis of symmetry: x = 1.52. Vertex: (1.5, -6.25 )3. Solutions: x = -1 or x = 44. y-intercept: (0, -5)

HW#1: Graphing Quadratic Functions

x

y

..

.(0, -5)

x = 4x = -1

x = 1.5

.(1.5, -6.25)

Given the below information, graph the quadratic function.

1. Axis of symmetry: x = 12. Vertex: (1, 0)3. Solutions: x = 1 (Double Root)4. y-intercept: (0, 2)

5. Hint: The axis of symmetry splits the parabola in half

x

y

.(1, 0)x = 1

x = 1

.(0, 2)

Calculating the Axis of Symmetry Algebraically

Ex: Find the axis of symmetry of y = x2 – 4x + 7

a = 1b = -4c = 7

a

bx

2

2)1(2

4

2

a

bx

2x

Calculating the Vertex AlgebraicallyEx1: Find the vertex of y = x2 – 4x + 7

a = 1, b = -4, c = 7

y = x2 – 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/2aStep 2: Substitute the x-value in the original function to find the

y-valueStep 3: Write the vertex as an ordered pair ( , )

2)1(2

4

2

a

bx

Ex3: (HW1 Prob #11)

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)

Vertex formula: Example: Find the vertex of y = 4x2 + 20x + 5

a = 4, b = 20, c = 5

y = 4x2 + 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/2aStep 2: Substitute the x-value in the original function to find the

y-valueStep 3: Write the vertex as an ordered pair ( , )

a

bx

2

5.2)4(2

20

2

a

bx

Ex4

Ex5

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)

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

2(x – 1)(x – 1) + 72(x2 – 2x + 1) + 72x2 – 4x + 2 + 72x2 – 4x + 9a = 2, b = -4, c = 9

y = 7 Answer: (1, 7)

1)2(2

4

2

a

bx

Graphing Quadratic Functions

For your given quadratic find the following algebraically (show all work!):

1. Find the axis of symmetry

2. The vertex

3. Find the solutions

4. Find the y-intercept

5. After you find the above, graph the quadratic on graph paper