8/14/2019 Quadratic Function 2

http://slidepdf.com/reader/full/quadratic-function-2 1/11

8/14/2019 Quadratic Function 2

http://slidepdf.com/reader/full/quadratic-function-2 2/11

All the slides in this presentation are timed.

You do not need to click the mouse or press any keys onthe keyboard for the presentation on each slide to continue.

However, in order to make sure the presentation does notgo too quickly, you will need to click the mouse or press a

key on the keyboard to advance to the next slide.

You will know when the slide is finished when you see a

small icon in the bottom left corner of the slide.

Click the mouse button to advance the slide when you see this icon.

8/14/2019 Quadratic Function 2

http://slidepdf.com/reader/full/quadratic-function-2 3/11

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

8/14/2019 Quadratic Function 2

http://slidepdf.com/reader/full/quadratic-function-2 4/11

 y = ax 2 + 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 Form

 y

 x

The standard form of a

quadratic function isa > 0

a < 0

8/14/2019 Quadratic Function 2

http://slidepdf.com/reader/full/quadratic-function-2 5/11

 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, wecould 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.

8/14/2019 Quadratic Function 2

http://slidepdf.com/reader/full/quadratic-function-2 6/11

Find the line of symmetry of 

 y = 3 x2 – 18 x + 7

Finding the Line of Symmetry

When a quadratic function is in

standard form

The equation of the line of symmetry is

 y = ax 2 + bx + c ,

2

b

a 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 3 x =

18

6= 3=

Thus, the line of symmetry is x = 3.

8/14/2019 Quadratic Function 2

http://slidepdf.com/reader/full/quadratic-function-2 7/11

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 theoriginal equation to find the y value.

 y = –2 x2 + 8 x –3

8 82

2 2( 2) 4

b

a 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)

8/14/2019 Quadratic Function 2

http://slidepdf.com/reader/full/quadratic-function-2 8/11

A Quadratic Function in StandardForm

The 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

2

b x

a

-=

8/14/2019 Quadratic Function 2

http://slidepdf.com/reader/full/quadratic-function-2 9/11

STEP 1: Find the line of 

symmetry

Let's Graph ONE! Try …

  y = 2 x2 – 4 x – 1

( )

41

2 2 2

b x

a

-= = =

A Quadratic Function in Standard Form

 y

 x

Thus the line of symmetry is x = 1

8/14/2019 Quadratic Function 2

http://slidepdf.com/reader/full/quadratic-function-2 10/11

Let's Graph ONE! Try …

  y = 2 x2 – 4 x – 1

STEP 2: Find the vertex

A Quadratic Function in Standard Form

 y

 x

( ) ( )2

2 1 4 1 1 3 y = - - =-

Thus the vertex is (1 ,–3).

Since the x – value of thevertex is given by the line of 

symmetry, we need to plug

in x = 1 to find the y – value

of the vertex.

8/14/2019 Quadratic Function 2

http://slidepdf.com/reader/full/quadratic-function-2 11/11

5 –1

Let's Graph ONE! Try …

  y = 2 x2 – 4 x – 1

( ) ( )2

2 3 4 3 1 5 y = - - =

STEP 3: Find two other points

and reflect them across the line

of symmetry. Then connect thefive points with a smooth

curve.

A Quadratic Function in Standard Form

 y

 x

( ) ( )2

2 2 4 2 1 1 y = - - =-

32

 y x