Vertex Form Calculator

If you want to find an equation's vertex form, feel free to use the vertex form calculator that evaluates the question quickly.

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What is the Vertex Form Calculator?

Vertex form Calculator with steps is a digital tool that is designed to help find the vertex from the given equation in less than a minute. Our tool evaluates the quadratic equations by converting them into the standard form of vertex form.

Vertex Form Calculator with Steps

The vertex calculator is specially made for students, educators, and anyone who wants to solve quadratic equations and the properties of quadratic functions quickly without any teacher.

What is Vertex Form?

Vertex form is a quadratic function that is a specific way to express a quadratic equation in a conic section. The vertex form converter uses the following form of vertex,

$$ f(x) \;=\; a(x - h)^2 + k $$

Where:

a is the coefficient that determines the direction of parabola along with magnitude of the opening ends. If a > 0, it opens upwards on the graph and if a < 0, it opens downwards.
(h,k) represents the coordinates of the vertex of the parabola.

The vertex is the point where the parabola reaches its maximum or minimum value, which depends on the direction of its opening. The x-coordinate of the vertex is given by h, while the y-coordinate is given by k.

How to Calculate Vertex Form?

To calculate the vertex form of a quadratic function from its standard form (ax² + bx + c), the vertex form calculator with steps finds the complete square of the given equation. Here are the steps to understand how to find the vertex form:

For example the vertex form of the quadratic function,

$$ f(x) \;=\; 2x^2 − 8x + 5 $$

Steps to Calculate Vertex Form:

First, write the quadratic equation in standard form equation such as:
$$ f(x) \;=\; ax^2 + bx + c $$

$$ f(x) \;=\; 2x^2 - 8x + 5 $$

Identify the coefficients: a = 2, b = −8, and c = 5.

Add and subtract the b in the given equation to make it a complete square

$$ cf(x) \;=\; a(x + 2ab)^2 − (4ab)^2 $$

$$ =\; 2(x^2 - 4x) + 5 $$

$$ =\; 2(x^2 - 4x + 4) + 5 + 4 $$

$$ =\; 2(x-2)^2 + 9 $$

Compare the above quadratic equation f(x) = 2(x-2)²+9 with vertex form equation.

$$ f(x) \;=\; a(x−h)^2 + k $$

$$ a(x−h)^2 + k \;=\; 2(x-2)^2 + 9 $$

Here,

$$ a \;=\; 2, (x-2)^2 \;=\; (x−h)^2 \;and\; k \;=\; 9 $$

So the given quadratic equation can be written in vertex form as,

$$ (h,k) \;=\; 2(x-2)^2 + 9 $$

How to Use Vertex Form Calculator?

The vertex form solver has a simple layout that enables you to calculate the vertex of the given quadratic equation. Follow our guidelines before using it to find the solution to the quadratic equation and to get an amazing experience. These guidelines are:

Enter the quadratic equation in the form of “ax² + bx + c” to find the vertex equation.
Review your equation before pressing the calculate button of the vertex form converter to get the accurate solution without any error.
Click on the “Calculate” button to get the solution of the quadratic equation problem.
Click on the “Recalculate” button of the vertex formula calculator to get a new page for more evaluation of the vertex form equation questions.
If you want to check the accuracy of our vertices calculator then try out the load example and get the solution.

Outcomes from Vertex Calculator:

The vertex form solver gives you a solution to your given quadratic equation problem after you click on the calculated button. It may include the following:

Result Option:

The result option provides you with solutions for vertex form equation problems.

Possible Steps:

Possible steps of the vertex formula calculator provide you with solutions to quadratic equation examples in a step-wise process.

Advantages of Vertex Form Converter:

The calculator for vertex form provides you many benefits whenever you use it to calculate the quadratic equation question to find its solution. These benefits help you to get a better understanding of vertex form concepts.

Vertex form grapher saves the time and energy that you consume to find the given equation questions manually.
It is a handy tool, so you can easily use it to solve the quadratic equation.
Vertex calculator has a user-friendly interface so that you can solve different equation vertexes easily.
The vertices calculator provides you with an exact solution as per your given input equation for the quadratic equation question.
Vertex Form Calculator with steps is a learning tool that helps you to become a pro with the concept of the vertex form equation with the help of the online platform.

Related References

How would you describe vertex form? Explain.
Give some examples of vertex form:
How to find the vertex form?
Explain vertex form of a quadratic function with the help of examples.

Frequently Ask Questions

What is h and k in vertex form?

In the vertex form of a quadratic function f(x) = a(x − h)² + k, the values of h and k denote the coordinates of the vertex of the parabola.

h represents the x-coordinate of the vertex.
k represents the y-coordinate of the vertex.

The vertex form f(x) = a(x−h)² + k provides a way to determine the vertex of the parabola without any graph of the function.

What is the vertex of a parabola in vertex form?

In the vertex form of a quadratic function f(x) = a(x−h)² + k, the vertex represents the point where the parabola reaches its maximum or minimum value, depending on its direction where it is opening. The equation of parabola is:

$$ y \;=\; x^2 $$

The vertex of the parabola is:

h is the x-coordinate of the vertex.
k is the y-coordinate of the vertex.

This means that the vertex form f(x) = a(x−h)² + k directly provides the coordinates of the vertex without making a graph of the function. For example the quadratic function,

$$ f(x) \;=\; 2(x − 3)^2 + 3 $$

Lets compare the given equation with vertex form f(x) = a(x - h)² + k.

As, $$ a(x - h)^2 + k \;=\; 2(x − 3)^2 + 3 $$

$$ h \;=\; 3,\; k \;=\; 2 $$

So, the vertex of the parabola represented by the function f(x) = 2(x − 3)² + 2 is at the point (h, k) as (3,2).

What is 2x^2-8x+9 in vertex form?

For the calculation of the quadratic function 2x² − 8x + 9 in vertex form, we follow the steps are

Identify the Coefficients of the given quadratic equation: a = 2, b = −8, and c = 9.
Convert the given equation into the standard form of the vertex equation.

$$ f(x) \;=\; 2x^2 − 8x + 9 $$

Add 4 on both sides

$$ =\; 2x^2 - 8x + 4 + 9 - 4 $$

$$ =\; 2(x-2)^2 + 1 $$

Calculate h and k by comparing them with the vertex form equation.

$$ f(x) \;=\; 2(x - 2)^2 + 1 $$

$$ f(x) \;=\; a(x - h)^2 + k $$

$$ a(x - h)^2 + k \;=\; 2(x - 2)^2 + 1 $$

$$ h \;=\; 2\; and\; k \;=\; 1 $$

So, the vertex form of the quadratic function 2(x - 2)² + 1 where the points (h, k) is (2, 1).

What is 3x^2-9x+10 in vertex form?

To express the quadratic function 3x² - 9x + 10 in vertex form as:

Identify the coefficients of the quadratic equation a = 3, b = −9, and c = 10.
Convert the given equation into the standard form of vertex equation.

$$ f(x) \;=\; 3x^2 - 9x + 10 $$

$$ f(x) \;=\; 3(x^2 - 3x) + 10 $$

$$ f(x) \;=\; 3(x^2 - 3x + \frac{9}{4}) + 10 + \frac{9}{4} $$

$$ f(x) \;=\; 3(\frac{x - 3}{2})^2 + \frac{49}{4} $$

$$ f(x) \;=\; 3(\frac{x - 3}{2})^2 + (\frac{7}{2})^2 $$

Calculate h and k by comparing them with the vertex form equation.

$$ a(x - h)^2 + k \;=\; 3(\frac{x - 3}{2})^2 + (\frac{7}{2})^2 $$

$$ h \;=\; \frac{3}{2},\; k \;=\; \frac{49}{4} $$

So, the vertex form of the quadratic function 3x² − 9x + 10 with points (h, k) as (3/2, 49/4).

What is f(x)=8x^2-4x written in vertex form?

To express the quadratic function f(x) = 8x² − 4x in vertex form, follow the steps as:

Identify Coefficients: a = 8, b = −4, and there is no constant term as c = 0.
Convert the given equation into the standard form of the vertex equation.

$$ f(x) \;=\; 8x^2 − 4x $$

$$ f(x) \;=\; 8(x^2 − \frac{1}{2x}) $$

$$ f(x) \;=\; 8(x^2 − \frac{1}{2x} + (\frac{1}{2})^2 + (\frac{1}{2})^2 $$

$$ f(x) \;=\; 8(\frac{x − 1}{2})^2 + \frac{1}{2} $$

Calculate h and k by comparing them with the vertex form equation.

$$ a(x - h)^2 + k \;=\; 8(\frac{x − 1}{2})^2 + \frac{1}{2} $$

$$ h \;=\; \frac{1}{4},\; k \;=\; \frac{1}{2},\; a \;=\; 8 $$

So, the vertex form of the quadratic function f(x) = 8x² − 4x is f(x) = 8(x − 1/2)² + ½ and points (h, k) are (¼, ½).

Amanda Jepson

Last Updated February 21, 2024