Vertex Form Calculator

Use our vertex form calculator that helps you to find the vertex of a parabola and the vertex form of a quadratic equation. This vertex to standard form calculator quickly displays vertex and y-intercept points with a graph.

Also, you can find how to find the vertex of a quadratic function, quadratic to vertex form, and vertex to standard form conversions in the context below.

What is the Vertex Form?

In the conic section, the vertex form of a parabola is a point or place where it turns, it is also known as a turning point. If the quadratic function converts to vertex form, then the vertex is (h, k).

The vertex equation is

$$ y = a(x – h)^2 + k $$

What is the vertex of a parabola?

“The point at the intersection of the parabola and its line is a symmetry known as the vertex of the parabola”.

How to Find the Vertex of a Parabola?

The vertex of a parabola is a specific point that represents the different values of the quadratic curve. The vertex can be either maximum (when parabola going downward) or minimum (when parabola going up). Therefore, the vertex form is the intersection of a parabola with its symmetric axis.

Normally, the vertex is (h, k), where h indicates the x-coordinates, and k stands for y-coordinates.

A standard form of a parabola $ ax^2 + bx + c $, so we can use quadratic equations of the vertex coordinates:

$$ h = -b / 2a $$

$$ k = c – b^2 / 4a $$

The above equations use our online standard to vertex form calculator to make calculations precisely.

However, an Online Parabola Calculator helps to find the standard form and vertex form of a parabola equation for the given values.

Example:

Finding the vertex of a parabola for the equation:

$$ = 2(x – (-6))^2 – 13 $$

Solution:

According to given equation

Vertex form is:

$$ y = 2 (x + 6)^2 – 13 $$

Standard form of given equation is:

$$ y = 2 x^2 + 24 x + 59 $$

Where,

Characteristic Points are:

Vertex P (-6, -13)

Y-intercept P (0, 59)

An online parabola vertex calculator can display a parabola graph with exact values when you substitute the same values for a vertex form equation.

How to Convert Standard form to Vertex form:

The standard form of a quadratic equation is $ m = a x^2 + b x + c $, where m and x are variables and a, b, and c are the coefficients. It is simple to solve an equation when it is in standard form because we calculate the answer with a, b, and c. However, when you need a graph of a parabola, quadratic function. The process is smooth when the equation is in vertex form. The standard to vertex form of a quadratic equation is $ Q = m(x – h)^2 + K $, where m represents the slope. Our standard form to vertex form calculator can change the standard to vertex form. Now get ready to know how to find vertex from standard form. Well, if you want to do it manually then follow these instructions:

Write the standard form of a quadratic function: $ m = a x^2 + b x + c $.
Divide first two terms by a: $ m = a (x^2 + b/a x) + c $.
Complete the square for the expression with x. Then, add and subtract $ (b/(2a))^2 $ from the equation: $ m = a [ x^2 + x (b/a) + (b/(2a))^2 – (b/(2a))^2] + c$.
Now, according to square formula, we can say write: $ m = a [(x + (b/(2a))^2 – (b/(2a))^2] + c $ and multiply the terms with a: $ m = a (x + (b/(2a))^2 – b^2/4a + c $.
Then compare the vertex equation: $ m = a (x – h)^2 + K $ , the vertex of parabola is: $ h = – b / 2a and k = c – b^2 / 4a $.

However, an Online Slope Calculator helps to find the slope (m) or gradient between two points in the Cartesian coordinate plane.

How to Convert vertex form to standard form:

A free online vertex form calculator can convert vertex form to the standard form of a parabola. If you want to know how to change the vertex to standard form, let’s start!

Write an equation in vertex form: $ m = a (x – h)^2 + K. $
Now, expand the square formula: $ m = a (x^2 + y^2 + 2hx) + K. $
Multiply the inner side or bracket: $ a x^2 + a y^2 + 2 ahx + K. $
Then, compare with quadratics in vertex form of a parabola: $ m = a x^2 + b x + c. $
Estimate the values of parameter: $ b = – 2 ah and c = a h^2 + k $.

How Vertex Form Calculator Works?

This vertex calculator can convert to vertex form or standard form with these steps:

Input:

First, select standard to vertex form or vertex form to standard form from the drop-down list.
Now, the vertex form of a parabola calculator displays an equation according to the selected option.
Then, substitute the value of variables according to the equation.
Click the calculate button to see the conversion and vertex points.

Output:

This vertex of a parabola calculator displays a vertex and standard form of the given equation.
This parabola to vertex form calculator also provides characteristic points with a parabola graph.

Reference:

From the source of Wikipedia: Etymology, Coefficients, Variables, The one-variable case, Bivariate case, Forms of a univariate quadratic function, Graph of the univariate function.