There are two general forms of the equation for a parabola.

Standard Form:

Vertex Form:

Using desmos graphing calculator, we can explore the parameter "a" in the form y = ax^2 +bx+c. With b =1 and c = 1, a ranging from values -10 to 10, we can see here what happens to the graph.

We learn that:

- a > 1 opens up, a < 1 opens down
- as |a| increases, the graph gets narrower
- as |a| decreases, the graph gets wider
- a = 0 is a linear equation
Using desmos graphing calculator, we can explore the parameter "b" in the form y = ax^2 +bx+c. With a = 1 and c = 1, b ranging from values -10 to 10, we can see here what happens to the graph.

We learn that:

- b shifts the vertex and the axis of symmetry
- axis of symmetry is x = -b/a (-b/a is also the x-coordinate of the vertex)
Now we can explore the parameter c. With a = 1 and b = 1, c ranging from values -10 to 10, we can see here what happens to the graph.

We learn that:

- c moves the graph up and down, vertical translation
- As c increases, the graph moves up
- As c decreases, the graph moves down
Let's now look at the vertex form of the parabola. This equation is y = a(x-h)^2 + k where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.

Using desmos graphing calculator, we see that the "a" in standard form and the "a" in vertex form are the exact same because it affects the graph in the same way. In the following graph, the values of a range from -10 to 10, h = 0 and k = 0. Click here to see what happens to the graph.

We learn that:

- a > 1 opens up, a < 1 opens down
- as |a| increases, the graph gets narrower
- as |a| decreases, the graph gets wider
- a = 0 is a linear equation
Using desmos graphing calculator, we can explore the affect h and k has on the graph. Click here to see the graph where a = 1, k = 0, and the values of h range from -10 to 10. Click here to see the graph where a = 1, h = 0, and the values of k range from -10 to 10.

We learn that:

- h changes the x-coordinate of the vertex of the parabola

- as h increases, the x-coordinate of the vertex increases
- as h decreases, the x-coordinate of the vertex decreases
- k changes the y-coordinate of the vertex of the parabola

- as k increases, the y-coordinate of the vertex increases
- as k decreases, the x-coordinate of the vertex decreases
Let's explore how to derive one formula from the other. We will start with the standard form of the parabola and end with the vertex form of the parabola by completing the square.

We have now derived the vertex formula of a parabola from the standard form.