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Graphing Quadratic Functions y = ax 2 + bx + c

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Page 1: Quadratic Equations   Graphing

Graphing Quadratic Functions

y = ax2 + bx + c

Page 2: Quadratic Equations   Graphing

Quadratic Functions

The graph of a quadratic function is a parabola.

A parabola can open up or down.

If the parabola opens up, the lowest point is called the vertex.

If the parabola opens down, the vertex is the highest point.

NOTE: if the parabola opened left or right it would not be a function!

y

x

Vertex

Vertex

Page 3: Quadratic Equations   Graphing

y = ax2 + bx + c

The parabola will open down when the a value is negative.

The parabola will open up when the a value is positive.

Standard Formy

x

The standard form of a quadratic function is

a > 0

a < 0

Page 4: Quadratic Equations   Graphing

y

x

Line of Symmetry

Line of Symmetry

Parabolas have a symmetric property to them.

If we drew a line down the middle of the parabola, we could fold the parabola in half.

We call this line the line of symmetry.

The line of symmetry ALWAYS passes through the vertex.

Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side.

Page 5: Quadratic Equations   Graphing

Find the line of symmetry of y = 3x2 – 18x + 7

Finding the Line of Symmetry

When a quadratic function is in standard form

The equation of the line of symmetry is

y = ax2 + bx + c,

2ba

x

For example…

Using the formula…

This is best read as …

the opposite of b divided by the quantity of 2 times a.

18

2 3x 18

6 3

Thus, the line of symmetry is x = 3.

Page 6: Quadratic Equations   Graphing

PracticeFind the Axis of Symmetry for each of the following Quadratic Functions

1. y = x2 – 4x + 3

2. y = x2 – 2x + 1

3. y = -x2 –2x + 3

4. y = -x2 + 4x – 3

5. y = x2 – 2x

6. y = -x2 + 2x

7. y = x2 – 1

Page 7: Quadratic Equations   Graphing

Practice - AnswersFind the Axis of Symmetry for each of the following Quadratic Functions

1. y = x2 – 4x + 3

2. y = x2 – 2x + 1

3. y = -x2 –2x + 3

4. y = -x2 + 4x – 3

5. y = x2 – 2x

6. y = -x2 + 2x

7. y = x2 – 1

1. x = 2

2. x = 1

3. x = -1

4. x = 2

5. x = 1

6. x = 1

7. x = 0

Page 8: Quadratic Equations   Graphing

Finding the Vertex

We know the line of symmetry always goes through the vertex.

Thus, the line of symmetry gives us the x – coordinate of the vertex.

To find the y – coordinate of the vertex, we need to plug the x – value into the original equation.

STEP 1: Find the line of symmetry

STEP 2: Plug the x – value into the original equation to find the y value.

y = –2x2 + 8x –3

8 8 22 2( 2) 4

ba

x

y = –2(2)2 + 8(2) –3

y = –2(4)+ 8(2) –3

y = –8+ 16 –3

y = 5

Therefore, the vertex is (2 , 5)

Page 9: Quadratic Equations   Graphing

PracticeFind the Vertex for each of the following Quadratic Functions

1. y = x2 – 4x + 3

2. y = x2 – 2x + 1

3. y = -x2 –2x + 3

4. y = -x2 + 4x – 3

5. y = x2 – 2x

6. y = -x2 + 2x

7. y = x2 – 1

Page 10: Quadratic Equations   Graphing

Practice - AnswersFind the Vertex for each of the following Quadratic Functions

1. y = x2 – 4x + 3

2. y = x2 – 2x + 1

3. y = -x2 –2x + 3

4. y = -x2 + 4x – 3

5. y = x2 – 2x

6. y = -x2 + 2x

7. y = x2 – 1

1. Vertex = (2, -1)

2. Vertex = (1, 0)

3. Vertex = (-1, 4)

4. Vertex = (2, 1)

5. Vertex = (1, -1)

6. Vertex = (1, 1)

7. Vertex = (0, -1)

Page 11: Quadratic Equations   Graphing

A Quadratic Function in Standard FormThe standard form of a quadratic function is given by

y = ax2 + bx + c

There are 3 steps to graphing a parabola in standard form.

STEP 1: Find the line of symmetry

STEP 2: Find the vertex

STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.

Plug in the line of symmetry (x – value) to

obtain the y – value of the vertex.

MAKE A TABLE

using x – values close to the line of symmetry.

USE the equation

2bxa

-=

Page 12: Quadratic Equations   Graphing

STEP 1: Find the line of symmetry

Let's Graph ONE! Try …

y = 2x2 – 4x – 1

( )4

12 2 2

bx

a

-= = =

A Quadratic Function in Standard Formy

x

Thus the line of symmetry is x = 1

Page 13: Quadratic Equations   Graphing

Let's Graph ONE! Try …

y = 2x2 – 4x – 1

STEP 2: Find the vertex

A Quadratic Function in Standard Formy

x

( ) ( )22 1 4 1 1 3y = - - =-

Thus the vertex is (1 ,–3).

Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex.

Page 14: Quadratic Equations   Graphing

5

–1

Let's Graph ONE! Try …

y = 2x2 – 4x – 1

( ) ( )22 3 4 3 1 5y = - - =

STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.

A Quadratic Function in Standard Formy

x

( ) ( )22 2 4 2 1 1y = - - =-

3

2

yx