How to find the vertex of a quadratic function calculator

An easy to use calculator to find the vertex, x and y intercepts of the graph of a quadratic function and write the function in vertex form.

f(x) = ax2 + bx + c

Vertex of the graph of a Parabola

The vertex of the graph of a parabola is the maximum or minimum point of the graph. This online calculator uses the formulas
h = - b / 2a
and
k = f(h)
to find the x and y coordinates h and k,respectively, of the vertex of a parabola.

Figure 1. Graph of a parabola with x (points A and B) and y (point C) intercepts and the vertex V.

x and y intercepts of the graph of a Parabola

To find the x intercepts, the calculator solves the quadratic equation ax2 + bx + c = 0 using the quadratic formulas:
x1 = (- b + √Δ) / (2 a)
x2 = (- b - √Δ) / (2 a)
where Δ = b2 - 4 a c is the discriminant.
The y intercept if given by f(0) = c.

Use of Calculator to Find Vertex and Intercepts of Quadratic Functions

1 - Enter the coefficients a, b, c as real numbers and the number decimal places as an integer and press "Solve". The coordinates of the x and y intercepts are displayed. If the two coordinates are equal, the graph touches the x axis and the two x intercepts have equal x coordinates. The x intercepts may not exist. The x and y coordinates of the vertex and the vertex form of the function are also displayed at the bottom.

Vertex Form

How to Use the Vertex Calculator?

The procedure to use the vertex calculator is as follows:

Step 1: Enter the values “a”, “b”, “c” in the respective input field

Step 2: Now click the button “Solve” to get the coordinates

Step 3: Finally, the vertex coordinates will be displayed in the output field

What is Meant by Vertex Coordinates?

In conic sections, the parabola vertex is a point where the parabola crosses its axis of symmetry. We know that the standard equation of a parabola is y = ax2+bx+c. If the coefficient of the term x2 is positive, the vertex will be in the bottom of the U- shaped curve. If the coefficient of the term x2 is negative, the vertex point will be in the top of the U-shaped curve. The standard equation of the parabola in terms of vertex equation is given by:

y = a(x-h)2 + k

Also, read: Conic Sections – Parabola

Example

Question:

Determine the coordinates of the vertex for the given parabola equation: y= 3x2+12x -12

Solution:

Given equation: y= 3x2+12x -12

Here a = 3, b = 12

We know that the formula to find the x- coordinate is given by -b/2a

= -12/2(3)

= -2.

Therefore, x -coordinate is -2

Now, substitute the value of x in the given equation, we get y= -24

Hence, the vertex coordinates (h, k) is (-2, -24)

An online vertex form calculator 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.

Let’s start with some basics!

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.

FAQ:

What is the vertex of an angle?

The vertex of an angle is the endpoint of two different rays that form the angle.

What is the common vertex?

A common vertex is shared by two angles. A vertex is a point of intersection where two linear construction lines intersect each other. However, the vertex form of a quadratic function calculator finds the common vertex of the parabola.

How we can find the turning point of a function?

A turning point of a line or function is a point where f′(x)=0. A turning point is a point where the parabola is upward (from decreasing to increasing) and f′(x)=0 at the point.

How we can determine the vertex with zeros?

First, find the zeros (0) by any factoring or the Quadratic Formula method. Now, find the x of the vertex by averaging the zeros. Then, we can calculate the f(x) to find out the y-coordinate of a vertex. Besides that, our online quadratic to vertex form calculator is the best way to calculate the vertex with zeros (0).

Final Thoughts:

Use this vertex form calculator to find the vertex and y-intercept points of the given equation. A special form of a quadratic function is a vertex form. With a parabola in vertex form calculator, we can see where the point of the parabola is maximum or minimum.

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.

From the source of Virtual Nerd: Vertex, Maximum and minimum points, Roots of the univariate function, Exact roots, Upper bound on the magnitude of the roots.

From the source of Math Bits Notebook: Iteration, Bivariate (two-variable) quadratic function, Minimum/maximum, Exact roots.

How do you find the vertex of the quadratic function?

We find the vertex of a quadratic equation with the following steps:.

Get the equation in the form y = ax2 + bx + c..

Calculate -b / 2a. This is the x-coordinate of the vertex..

To find the y-coordinate of the vertex, simply plug the value of -b / 2a into the equation for x and solve for y..

What is the vertex of a parabola calculator?

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.