Finding the Vertex Using the X-intercepts

Finding the Vertex from Roots

You now know how to find the x-intercepts from factoring. Now we will use the x-intercepts to find the vertex.

We know that the shape of a parabola is symetrical. What divides the parabola? The axis of symmetry.

Finding the Axis of Symmetry

or

Line of Symmetry.

It is a line so it will always look like x = #.

Multiple Choice

The line that goes through the middle of a parabola is called the...

• Vertex

• Axis of Symmetry

• x-intercept

• y-intercept

Multiple Choice

What is the green dashed line called?

• roots or x-intercepts
• parabola
• axis of symmetry
• line of dashes

Did you notice that the axis of symmetry went through the vertex?

Multiple Choice

What is the equation for axis of symmetry?

• y=3
• x=3
• x=5
• x=1

The axis of symmetry was x = 3.

We will use that 3 as the x value in our VERTEX!

Multiple Choice

What is the vertex of this parabola?

• (0, 5)

• (0, 0)

• (-4, 5)

• (-2, 1)

Multiple Choice

What is the axis of symmetry?

• y = 5

• y = 1

• -2

• x = -2

Multiple Choice

What is the vertex?

• (4,-4)
• (2,0)
• (0,2)
• (6,0)

Multiple Choice

What is the axis of symmetry?

• x = 4

• x = 2

• x = 6

• (4, -6)

Finding the axis of symmetry from the x-intercepts.

If you noticed from the last slide, the x-intercepts were 2 and 6, the axis of symmetry was x = 4. Do you see how the intercepts are related to the axis of symmetry? What is between 2 and 6? 4!

If you can find the x-intercepts you can find the axis of symmetry.

Find the average of the x-intercepts by adding them together and dividing them by 2.
Example

 $\frac{\left(2+6\right)}{2}\ =\ \frac{8}{2}\ =\ 4$  
The axis of symmetry is x = 4

Multiple Select

What are the x-intercepts?

• -3

• 5

• 8

• 7 1/2

Multiple Choice

What is the axis of symmetry?

• x = -3

• x = 5

• x = 1

• x = 8

The average of -3 and 5 is 1

 $\frac{-3\ +\ 5}{2}\ =\frac{2}{2}\ =\ 1$ 

 

x = 1 is the axis of symmetry

Poll

How do you feel about finding the axis of symmetry?

• I got this!

• With some practice I'll be good.

• Whoa, I'm going to need a LOT of practice.

• An axis of what??

Now that we can find the axis of symmetry

We can also find the vertex.

How?

Plug in the value of the axis of symmetry in for x and solve for y!

Multiple Choice

Find the zeros of the equation: 
(x+10)(x-2) = 0

• 10 and 2
• -2 and 10
• -10 and 2
• -2 and -10

Now you know that the

zeros, roots, or x-intercepts

are -10 and 2, find the axis of symmetry

 $\frac{-10+2}{2}=\frac{-8}{2}=\ -4$  

Plug in -4 for x and solve for y
y = (x + 10)(x - 2)
y = (-4 +10)(-4 - 2)
y = (6)(-6)
y = -36
The vertex is (-4, -36)

What are the x-intercepts? 2 and 6

The average of 2 and 6 is 4
plug 4 in to equation

 $y\ =\ x^{2\ }-8x\ +12$  

 $y\ =\ 4^2\ -8\left(4\right)\ +\ 12$   $y\ =\ 16\ -\ 32\ +\ 12$  
 $y\ =\ -4$  
Vertex is (4, -4)  Whew!

Multiple Choice

What is the vertex?

• (4,-4)
• (2,0)
• (0,2)
• (6,0)

You Try!!

Multiple Choice

What are the zeros of the parabola? 

• 1, 5
• 3, -5 
• 3, 0
• -6, 1

Multiple Choice

What is the axis of symmetry?

• x = 2

• x = 3

• x = 5

• x = 1

Multiple Choice

If the factored form of this graph is 

 $y\ =\ \left(x-1\right)\left(x-5\right)$ 
What would your vertex be?

• (3, 32)

• (3, -4)

• (3, -2)

Finding the Vertex from the X-intercepts

• Factor and solve to find the x-intercepts

• Find the axis of symmetry by finding the average of the x-intercepts

• Plug the value of the axis of symmetry in for x and solve for y.

• Your (x, y) is your vertex

• I DID IT!!!

Multiple Choice

Factor the following quadratic expression:

x² + 10x + 24

• (x + 6)(x + 4)

• (x + 10)(x + 2)

• (x + 8)(x + 3)

• (x + 2)(x + 12)

Multiple Select

If your factors were (x+6)(x+4), find the x-intercepts (choose 2):

x² + 10x + 24 = 0

• x = -6

• x = -8

• x = -4

• x = -3

Multiple Choice

If the x-intercepts for this equation are -6 and -4 what is the axis of symmetry?

x² + 10x + 24 = 0

• x = -10

• x = -5

• x = 10

• x = 8

Multiple Choice

If the axis of symmetry is x = -5, what is the vertex?

x² + 10x + 24 = 0

• (-5, -51)

• (-5, 84)

• (-5, -1)

• (-5, -36)

Did you get it right??

Practice makes perfect.

Follow the steps and you will get there.

Poll

I know how to find the axis of symmetry and the vertex for a quadratic using the x-intercepts.

• 100% True

• 75% True

• 50% True

• Not yet BUT I WILL!!!