Graphs of Quadratics

Let’s start by graphing the parent quadratic function

y = x2

Graphs of Quadratics

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

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

a, b, and c are the coefficientsExample:

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

Characteristics of Quadratic Functions When the power of an equation is 2, then the

function is called a quadratic function 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

In general equations have roots,

Functions haves zeros, and

Graphs of functions have 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)

maximum

Axis of symmetry examples

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

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)

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)

Hint: The axis of symmetry splits the parabola in half

x

y

.(1, 0)x = 1

x = 1

.(0, 2)

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?5. What is the domain?6. What is the range?

Ex: Graph 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

5. What is the domain? All real numbers

6. What is the range? y ≥ -4

Finding the y-intercept

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

Answer: c

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 (HW1 Prob #9)

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)

Find the vertex: y = 2(x – 1)2 + 72(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

(HW1 Prob #12)

SWBAT… graph quadratic functions. Mon, 5/21

Agenda 1. WU (15 min)2. Graphs of quadratic functions - posters (30 min)

Warm-Up:

1. Take out HW#1: Any questions?

2. Review the weekly agenda

HW#2: Quadratic functions (both sides)

HW1, Problem #4

Axis of symmetry: x = -2 Vertex: (-2, -1) y-intercept: (0, 3) Solutions: x = -3 or x = -1 Domain: All real numbers Range: y ≥ -1

Graph y = -x2 + 1 (HW1 Prob #2)

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

5. What is the domain?

6. What is the range?

All real numbers

y ≤ 1

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 (HW1 Prob #9)

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)

Find the vertex: y = 2(x – 1)2 + 72(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

(HW1 Prob #12)

Graphing Quadratic Functions For your given quadratic find the following

algebraically (show all work on poster!):1. Find the axis of symmetry

2. The vertex (ordered pair)

3. Find the solutions

4. Find the y-intercept (ordered pair)

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

6. Find the domain

7. Find the range (need the vertex!)

Exit Slip: Complete on graph paper:

Given y = x2 + 6x + 8 find algebraically:1. The axis of symmetry2. The vertex (as an ordered pair)3. The solutions (x-intercepts)4. The y-intercept (as an ordered pair)5. After you find the above, graph the quadratic 6. Domain7. Range