Free Vertex Form Calculator

The vertex form calculator that helps you to find the vertex of a parabola and the vertex form of a quadratic equation. With that, the calculator quickly displays vertex and y-intercept points with a graph.

What Is the Vertex Form?

The vertex form of a parabola is a point or place where it turns. 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:

\(h = -b / 2a\)

\(k = c – b^2 / 4a\)

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:

\(Vertex = P (-6, -13)\)

\(Y-intercept = P (0, 59)\)

How To Convert Standard Form To Vertex Form?

The standard form of a quadratic equation is \(ax^2 + bx + c=0\), 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.

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. If you want to get vertex from the standard form, follow these points:

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: \( h = – b / 2a and k = c – b^2 / 4a \).

How To Convert Vertex Form To Standard Form?

This tool 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: \( b = – 2 ah and c = a h^2 + k \).

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.