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

• $$y=x^2+5x$$

• $$y=3\left(x+4\right)^2-1$$

• $$y=-\left(x+3\right)^2-2$$

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

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