Vertex Form

Vertex Form of a Quadratic Equation

Forms of a Quadratic Equation

• Standard Form

• Vertex Form

• Factored Form

Standard Form

• Standard Form of a quadratic is: y = ax²+bx+c.

• a,b,c are coefficients. They are just numbers.

• We have sees some examples before like:

• x²+4x+3

• x²+1

• 2x²-5x+6

Standard Form

• The c value is the y-intercept

• To find the equation of the axis of symmetry, you can use:

• x = -b/2a

• If the a value is negative, the parabola is opening down.

if a is negative it opens down

If a is positive, it opens up

Vertex Form

• Vertex form is a form of a quadratic that allows us to read off the vertex just from looking at the equation.

• Vertex form is:

•  $y=a\left(x-h\right)^2+k$  

• (h,k) are the coordinates of your vertex. 

• a is the same a value from standard form. It represents if the parabola is stretched or compressed. 

Multiple Select

Check all of the equations that are in standard form

• $$y=2x+1$$

• $$y=x^2+4x-9$$

• $$y=4\left(x-5\right)^2+3$$

• $$y=x^2+9$$

Multiple Select

Select all equations that are in vertex form

• $$y=x^2+5x$$

• $$y=3\left(x+4\right)^2-1$$

• $$y=-\left(x+3\right)^2-2$$

• $$y=\left(x+3\right)\left(x-2\right)$$

Fill in the Blanks

Standard form gives you the

Fill in the Blanks

Vertex form gives you the

Vertex Form

Looking at the graph on the right, the vertex is (1,-4). If a=1, how can we write this in vertex form?

Vertex Form

Vertex form is

 $y=a\left(x-h\right)^2+k$  
The vertex is (1,-4). We consider the x-coordinate to be h and the y-coordinate to be k.

I mentioned that a=1

Vertex Form

Vertex form is

 $y=a\left(x-h\right)^2+k$  
Let's substitute those values in:

 $y=1\left(x-1\right)^2-4$  

Vertex Form

This quadratic has an a value of 2

The vertex is (-3,1)

Vertex Form

Let's substitute these values in: 

 $y=a\left(x-h\right)^2+k$  
 $y=2\left(x-\left(-3\right)\right)^2+1$  
 $y=2\left(x+3\right)^2+1$  

Vertex Form

 $y=2\left(x+3\right)^2+1$  
Notice how the vertex is (-3,1), and when substituted into the equation, it becomes a +

Multiple Choice

If given the equation y = 3(x + 5)² - 4, what is the vertex of the parabola?

• (5, -4)
• (-5, -4)
• (-15, -4)
• (15, -4)

Multiple Choice

What is the equation of this graph?

• f(x) = (x -5)² + 1
• f(x) = (x - 1)² - 5
• f(x) = (x + 1)² - 5
• f(x) = (x + 5)² +1

Multiple Choice

If given the equation
y = 3(x + 5)² - 4,
what is the vertex of the parabola?

• (5, -4)
• (-5, -4)
• (-15, -4)
• (15, -4)

Vertex Form

There are a few other interesting things we can get from vertex form

1. The a value will still determine if the parabola opens up or down
2. We can get the equation of the axis of symmetry, since the axis of symmetry goes through the x-coordinate of the vertex

Writing an equation in Vertex Form

Example:A quadratic is stretched by a value of 2. The vertex is (1,3). It is opening up. $y=a\left(x-h\right)^2+k$  

 $y=2\left(x-1\right)^2+3$  

Writing an equation in Vertex Form

Example:A quadratic is stretched by a value of 1/2. The vertex is (-1,-3). It is opening down. $y=a\left(x-h\right)^2+k$  

 $y=\frac{1}{2}\left(x+1\right)^2-3$  

Multiple Choice

The parabola y = - 2x² - 4x + 1 will have an axis of symmetry of

• x = - 1

• x = 1

• x = 4

• x = - 4

Multiple Choice

The parabola y = x² - 2x + 4 will

• open up

• open down

Multiple Choice

What is the vertex?

• (4,-4)
• (2,0)
• (0,2)
• (6,0)

Multiple Choice

Does the parabola
f(x)=4(x-2)² + 3
open up or down?

• up
• down

Multiple Choice

What is the vertex of the parabola: y=2(x-3)²+4

• (3,4)
• (-3,4)
• (3, -4)
• (-3,-4)

Multiple Choice

A parabola has a vertex at (-3,2).
Where is the axis of symmetry?

• y = -2
• x = 3
• x = -3
• y = 2

Multiple Choice

What is the green dashed line called?

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

Multiple Choice

The axis of symmetry line passes through which point?

• y-intercept
• any random point
• vertex
• symmetrical point for the vertex

Open Ended

Briefly explain what information standard form of a quadratic can give you

Open Ended

Briefly explain what information vertex form of a quadratic can give you