# Parabola Graphing Calculator

### What is the purpose of the Parabola Graphing Calculator?

It allows you to graph Parabolas. In particular, you can experiment with the 3 different forms of Quadratic Equations: Vertex, Standard and Factored Form.
Different equations may produce the same Parabolas.

For example, these 3 equations will graph the same Parabola. Try it below.
y = (x + 3)2 -1
y = x2 + 6x + 8
y = (x+4)*(x+2)

# Parabola Graphing Calculator

Click on the blue, green and purple circles below to disable the graphs. Then click again – one circle at a time – to show one Graph at a time. You will see that 3 quadratic equations will form the same Parabola.

Can you create 3 different equations that result in the same Parabola?
Try it above by typing in 3 different equations into the Parabola Graphing Calculator above.
Hint: All 3 equations would be equal when converted to Standard Form.

Read below on how to convert between the 3 Parabola forms.

### Example of the 3 different forms?

Standard Form:
 y = x²+6x+8
can be rewritten in Vertex Form:
 y = (x+3)²-1
It tells us that above Parabola has Vertex Coordinates = (3,-1). Verify that above on the Graph of the Parabola Graphing Calculator. You can actually click on the vertex and the coordinates will show.

The process to convert from Standard Form to Vertex Form is called “Completing the Square”. You can Complete the Square for your quadratic equation below.

You can read the details and examples on how to convert here.

Factored Form:
 y = (x+4)(x+2)
It tells us that above Parabola has Zeros -4 and -2 . Again, verify that above on the Graph of the Parabola Graphing Calculator. You can actually click on the zeros and the coordinates will show.

The process to convert from Standard Form to Factored Form can be done below.

You can read the details and examples on how to convert here.

Or simply use the head menu when converting between the different Parabola Forms.

### Summary: The 3 different Forms of Quadratic Equations

The 3 different forms of Quadratic Equations:

Standard Form:
 y = ax²+bx+c
Vertex Form:
 y = a(x-h)²+k
(h,k) = Vertex Coordinates.

Factored Form:
 y = a(x-r)(x-s)
r and s = Zeros of the Parabola.

### Calculate the Parabola Equation

When given the Vertex and a Point you will use the Vertex Form: y = a(x-h)²+k
(h,k) = Vertex Coordinates.
Example: Vertex=(1,2) which makes the Parabola Equation y = a(x-1)²+2 .
Given Point (3,10) turns the Equation to 10 = a(3-1)²+2 = 4*a+2 implying that a=2.
Therefore, the Parabola Equation is calculated as y = 2(x-1)²+2

If however you are given two Zeros and a Point you will use the Factored Form: y = a(x-r)*(x-s)
Example: Zeros are 3 and 5 which makes the Parabola Equation y = a(x-3)*(x-5) .
Given Point (4,-2) makes the Parabola Equation -2 = a(4-3)*(4-5) = a*(-1) = -a which implies a=2.
Therefore, the Parabola Equation is calculated as y = 2(x-3)*(x-5).