Vertex of a Parabola Calculator

What is a Parabola?

A parabola is a special U-shaped curve. Imagine throwing a ball into the air - the path it follows forms a parabola! You can also see parabolas in the shape of satellite dishes or the arches of some bridges.

How to Calculate a Parabola

To find a parabola, we use a special equation. It's like following a recipe to make your favorite snack, but instead of ingredients, we use numbers and math symbols to create our parabola shape.

Formula

There are two common forms of a parabola equation:

Vertex Form:

\[ y = a(x-h)^2 + k \]

This means:

• \((h,k)\) is the vertex of the parabola (the bottom or top of the U-shape)
• \(a\) tells us how wide or narrow the parabola is, and which way it opens
• \(x\) and \(y\) are the coordinates of any point on the parabola

Standard Form:

\[ y = ax^2 + bx + c \]

Where:

• \(a\), \(b\), and \(c\) are constants, and \(a \neq 0\)
• \(a\) determines the direction and width of the parabola
• \(b\) and \(c\) affect the position of the parabola

Calculation Steps

1. Find the vertex (h,k) of the parabola
2. Determine the value of a
3. Write the equation using these values
4. Use the equation to find points on the parabola
5. Plot these points to draw the parabola

Example and Visual Representation

Let's look at a parabola with vertex (2,1) and a=0.5.

The equation in vertex form would be: \(y = 0.5(x-2)^2 + 1\)

The same equation in standard form would be: \(y = 0.5x^2 - 2x + 3\)

Here's what this parabola looks like:

In this picture, you can see the U-shape of the parabola. The vertex is at (2,1), which is the lowest point of the curve. The parabola opens upward because a is positive (0.5).