how to find standard form of a parabola

Parabola Calculator

Created past Bogna Szyk and Wojciech Sas , PhD candidate

Reviewed past

Steven Wooding and Jack Bowater

Last updated:

Jan xviii, 2022

Table of contents:

• What is a parabola?
• The parabola equation in vertex course
• Parabola focus and directrix
• How to use the parabola equation calculator: an example
• FAQ

Any time yous run into a quadratic formula you want to analyze, you'll find this parabola estimator to be the perfect tool for y'all. Not only volition it provide you with the parabola equation in both the standard grade and the vertex form, but also calculate the parabola vertex, focus, and directrix for you.

What is a parabola?

source: Wikimedia

A parabola is a U-shaped symmetrical curve. Its chief belongings is that every indicate 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 ever 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 plough; it lies halfway between the focus and the directrix.

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

The parabola equation in vertex course

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

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

• a — Same equally the a coefficient in the standard grade;
• h — x-coordinate of the parabola vertex; and
• one thousand — y-coordinate of the parabola vertex.

You tin can summate the values of h and thousand from the equations below:

h = - b/(2a) k = c - b²/(4a)

Parabola focus and directrix

The parabola vertex form calculator also finds the focus and directrix of the parabola. All you have to do is to utilize the post-obit equations:

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

How to apply the parabola equation calculator: an example

1. Enter the coefficients a, b and c of the standard course of your quadratic equation. Let's assume that the equation is y = 2x² + 3x - 4, what means that a = 2, b = 3 and c = -4.

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

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

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

3. Notice the coordinates of the focus of the parabola. The ten-coordinate of the focus is the same as the vertex's (10₀ = -0.75), and the y-coordinate is:

   y₀ = c - (b² - 1)/(4a) = -4 - (nine-1)/8 = -v

4. Find the directrix of the parabola. You can either use the parabola estimator to do it for you, or y'all can utilise the equation:

   y = c - (b² + ane)/(4a) = -4 - (ix+i)/8 = -five.25

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

FAQ

What is a parabola?

A parabola is a symmetrical U shaped curve such that every point on the curve is equidistant from the directrix and the focus.

How do I define a parabola?

A parabola is defined by the equation such that every bespeak on the curve satisfies information technology. Mathematically, y = ax² + bx + c.

How do I summate the vertex of a parabola?

To calculate the vertex of a parabola defined by coordinates (x, y):

1. Notice x coordinate using the axis of symmetry formula:

   x₀ = - b/(2a)

2. Find y coordinate using the equation of parabola:

   y₀ = c - (b² - one)/(4a)

How to calculate the focus of a parabola?

To calculate the focus of a parabola defined by coordinates (x, y):

1. Find y coordinate using the formula y = c - (b² + 1)/(4a)
2. Find x coordinate using the parabola equation.

Bogna Szyk and Wojciech Sas , PhD candidate

What to input?

Standard form: y = ax² + bx + c

Results

Show results using fractions?

Average rate of modify Bilinear interpolation Catenary curve … 35 more

Source: https://www.omnicalculator.com/math/parabola#:~:text=The%20standard%20form%20of%20a,parabola%20%E2%80%93%20its%20vertex%20and%20focus.

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