Parabola Calculator

A parabola is a U-shaped symmetrical curve. Its main property is that every point lying on the parabola is equidistant from both a certain point, called the focus of a parabola, and a line, called its directrix. It is also the curve that corresponds to quadratic equations.

The axis of symmetry of a parabola is always perpendicular to the directrix and goes through the focus point. The vertex of a parabola is the point at which the parabola makes its sharpest turn; it lies halfway between the focus and the directrix.

A real-life example of a parabola is the path traced by an object in projectile motion.

The standard form of a quadratic equation is y = ax² + bx + c. You can use this vertex calculator to transform that equation into the vertex form, which allows you to find the important points of the parabola – its vertex and focus.

The parabola equation in its vertex form is y = a(x - h)² + k, where:

• a — Same as the a coefficient in the standard form;
• h — x-coordinate of the parabola vertex; and
• k — y-coordinate of the parabola vertex.

You can calculate the values of h and k from the equations below:

h = - b/(2a)

k = c - b²/(4a)

The parabola vertex form calculator also finds the focus and directrix of the parabola. All you have to do is to use the following equations:

• Focus x-coordinate: x₀ = - b/(2a);
• Focus y-coordinate: y₀ = c - (b² - 1)/(4a); and
• Directrix equation: y = c - (b² + 1)/(4a).

Let's assume that the equation is y = 2x² + 3x - 4. To use parabola calculator, follow these steps:

1. Select one of the options in the “What to input” list. In our case, it's the Standard form.

2. Decide on the parabola's orientation. It's vertical (y 〜 x²).

3. Enter the coefficients a, b, and c of the standard form of your quadratic equation. These are: a = 2, b = 3 and c = -4.

4. And that's all! All the remaining characteristics of the parabola will show up in the Result section.

We can also evaluate some of them by hand:

1. Calculate the coordinates of the vertex, using the formulas listed above:

   h = - b/(2a) = -3/4 = -0.75

   k = c - b²/(4a) = -4 - 9/8 = -5.125

2. Find the coordinates of the focus of the parabola. The x-coordinate of the focus is the same as the vertex's (x₀ = -0.75), and the y-coordinate is:

   y₀ = c - (b² - 1)/(4a) = -4 - (9-1)/8 = -5

3. Work out the directrix of the parabola. You can either use the parabola calculator to do it for you, or you can use the equation:

   y = c - (b² + 1)/(4a) = -4 - (9+1)/8 = -5.25

If you want to learn more coordinate geometry concepts, we recommend checking the average rate of change calculator and the latus rectum calculator.