Different forms of a quadratic?

What are the different forms of a quadratic equation?

Intro to vertex form.

Multiple Select

How many equivalent ways can we write a quadratic equation?


 $f\left(x\right)=ax^2+bx+c$  

• 1 

• 2 

• 3

• I'm not sure.

Multiple Select

 $f\left(x\right)=x^2-6x+8$  
This quadratic is equivalent to ...

•  $f\left(x\right)=\left(x-2\right)\left(x-4\right)$  

•  $f\left(x\right)=\left(x-3\right)^2-1$  

Poll

We write quadratics in different forms. 
We have the standard form like:
 $f\left(x\right)=x^2-6x+8$  
We have the factored form like:
 $f\left(x\right)=\left(x-2\right)\left(x-4\right)$ 
And this one:  $f\left(x\right)=\left(x-3\right)^2-1$   

Which we call... 

• Super Saiyan Form

• Nine-Tails Chakra Form

• Vertex Form

• Tenth Form: Constant Flux/The Dragon of Change.

Multiple Choice

Why is it that we call quadratics written like this
 $y=3\left(x+2\right)^2-27$  

the vertex form?

• Because it highlights the coordinates of the vertex of the function's graph.

• Because it highlights the line of symmetry of the function's graph.

• Because it highlights the x-intercepts of the function's graph.

Poll

 $y=-2\left(x-3\right)^2-4$  has a vertex at...

•  $\left(-3,4\right)$  

•  $\left(3,-4\right)$  

•  $\left(1,-12\right)$  

Multiple Choice

 $y=-2\left(x-4\right)^2+1$  

What is the vertex of the quadratic:

•  $\left(1,\ 4\right)$  

•  $\left(1,\ -4\right)$  

•  $\left(4,\ 1\right)$  

•  $\left(-4,\ 1\right)$  

Multiple Choice

 $y=\left(x-4\right)^2+2$  

What is the vertex of the quadratic:

•  $\left(2,\ 4\right)$  

•  $\left(4,\ 2\right)$  

•  $\left(-4,\ -2\right)$  

Poll

How confident are you so far at identifying the vertex?

• Very confident

• Pretty good

• Yikes!

Multiple Choice

 $f\left(x\right)=2\left(x-1\right)^2-4$  

What is the vertex of the quadratic?

•  $\left(1,\ -4\right)$  

•  $\left(-1,\ -4\right)$  

Multiple Choice

 $f\left(x\right)=2\left(x+2\right)^2+4$  

What is the vertex of the parabola?

•  $\left(2,\ 4\right)$  

•  $\left(-2\ ,\ 4\right)$  

Multiple Choice

 $f\left(x\right)=-2\left(x+4\right)^2-1$  

What is the vertex of the parabola?

•  $\left(\ 4,\ -1\right)$  

•  $\left(-4,\ -1\right)$  

•  $\left(-4,\ +1\right)$  

Multiple Select

The path of the rocket is given by the equation:

 $y=-16\left(x-4\right)^2+256$  
Where the height h  is in feet of a small rocket x seconds after it is launched.

 What is the greatest height the rocket reaches?

• 4 feet in 256 seconds (4,  256)

• 256 feet after 4 seconds. (4, 256)

Poll

Which form of the quadratic equation (parabola) do you find interesting?

• Standard Form

  
  $y=ax^2+bx+c$
  

• Vertex Form
  $y=a\left(x-h\right)^2+k$
  

• Factored Form
  $y=a\left(x-b\right)\left(x-c\right)$
  

What is the benefit of the factored form?

What does it tell you? What does it leave out?

Okay!

Let's check out what we learned!

Multiple Choice

 $y=\left(x+5\right)\left(x+1\right)$  

What does this quadratic form tell you?

• The vertex is at  $\left(-5,\ -1\right)$  

• The y-intercept is at  $\left(0,\ 5\right)$  

• The x-intercepts are  $\left(-5,\ 0\right)\ and\ \left(-1,\ 0\right)$  

Multiple Choice

 $y=x^2+x-30$  

What does this quadratic form tell you?

• The vertex is at  $\left(0,\ 30\right)$  

• The y-intercept is at  $\left(0,\ -30\right)$  

• The x-intercepts are  $\left(30,\ 0\right)\ and\ \left(-30,\ 0\right)$  

Multiple Choice

 $y=\left(x-3\right)^2+2$  

What does this quadratic form tell you?

• The vertex is at  $\left(3,\ 2\right)$  

• The y-intercept is at  $\left(0,\ 2\right)$  

• The x-intercepts are  $\left(3,\ 0\right)\ and\ \left(2,\ 0\right)$  

Multiple Choice

 $y=\left(x+1\right)\left(x-6\right)$  

What does this quadratic form tell you?

• The vertex is at  $\left(-1,\ 6\right)$  

• The y-intercept is at  $\left(-1,\ 6\right)$  

• The x-intercepts are  $\left(-1,\ 0\right)\ and\ \left(6,\ 0\right)$  

Poll

This is how I feel about the different forms of Quadratics!