A Vertex Calculator is a free online tool that displays a vertex of a given expression. STUDYQUERIES’S online vertex calculator tool makes the calculations faster and easier where it shows the vertex point in a fraction of seconds.

How to Use a Vertex Calculator?

The procedure to use a vertex calculator is as follows. There are two approaches you can take to use our vertex form calculator:

  1. The first possibility is to use the vertex form of a quadratic equation;
  2. The second option finds the solution of switching from the standard form to the vertex form.

We’ve already described the last one in one of the previous sections. Let’s see what happens for the first one:

  • Type the values of parameter a, and the coordinates of the vertex, \(h\), and \(k\). Let them be \(a = 0.25,\ h = -17,\ k = -54\);
  • That’s all! As a result, you can see a graph of your quadratic function, together with the points indicating the vertex, y-intercept, and zeros.

Below the chart, you can find the detailed descriptions:

  • Both the vertex and standard form of the parabola: \(y = 0.25(x + 17)² – 54\) and \(y = 0.25x² + 8.5x + 18.25\) respectively;
  • The vertex: \(P = (-17, -54);\)
  • The y-intercept: \(Y = (0, 18.25);\)
  • The values of the zeros: X₁ = (-31.6969 , 0), X₂ = (-2.3031, 0). In case you’re curious, we round the outcome to five significant figures here.

Vertex Calculator

What Is Vertex: Definition

Vertex is the point where two or more lines or edges meet to form an angle. In this lesson, you will learn about vertex examples, the vertex definition, and the vertex angle.

Vertex- Definition
Vertex- Definition

Let’s have a look at the two rays: Ray 1 and Ray 2, which originate from the vertex O.

Acute angle geometry: When two rays meet and an angle is formed that is greater than 0 degrees and less than 90 degrees, it is called an acute angle.

As we go forward, you can check out the interactive examples to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

Learn all you need to know about vertex in this short article!

What Is a Vertex In Triangle?

A vertex in math is a point where two lines or rays meet. An angle is formed at the vertex. A vertex is denoted by uppercase letters like A, O, P, etc.

What Is Vertex In Triangle
What Is Vertex In Triangle

The plural of vertex is “vertices.” In solid geometry, i.e., three-dimensional geometry, shapes such as cubes, cuboids form several vertices. Let us look at them in detail in the next section.

Look at the triangle DEF.

it has three vertices: D, E, and F that form the angles of the triangle.

What Is a Vertex In Solid Geometry?

Not just plane shapes, even solid shapes have vertices. In solid shapes, a vertex is formed where edges meet.

What Is a Vertex In Solid Geometry
What Is a Vertex In Solid Geometry

Look at this cube,

A, B, C, D, E, F, G, H are the vertices of the cube.

A tetrahedron has 4 vertices. One is marked. Can you identify the other 3 vertices of the tetrahedron?

A Tetrahedron Has 4 Vertices
A Tetrahedron Has 4 Vertices

We can apply Euler’s formula to find the vertices in a solid shape.

Euler’s formula is: $$F + V − E = 2$$


  • F is the number of faces
  • V stands for the vertices
  • E is the number of edges

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Vertex Of Parabola In Math

In Mathematics, the vertex formula helps to find the vertex coordinate of a parabola, when the graph crosses its axes of symmetry. Generally, the vertex point is represented by \((h, k)\). We know that the standard equation of a parabola is $$y=ax^2+bx+c$$ Here, if the coefficient of \(x^2\) is positive, the vertex should be at the bottom of the U-shaped curve. If the coefficient of \(x^2\) is negative, then the vertex should be at the top of the U-shaped curve. In this article, we are going to learn the standard form and vertex form of a parabola, the vertex formula, and examples in detail.

Vertex Form of Parabola

We know that the standard form of the parabola is $$y=ax^2+bx+c$$

Thus, the vertex form of a parabola is $$y = a(x-h)^2 + k$$

Now, let us discuss the vertex formula in detail.

Vertex Of Parabola Graph Representation In Math
Vertex Of Parabola Graph Representation In Math

Vertex Formula

The vertex formula is used to find the vertex of a parabola. There are two ways to find the vertex of a parabola.

$$Vertex,\ (h, k) = (\frac{-b}{2a}, \frac{-D}{4a})$$

Where “D” is the discriminant where $$D = b^2 – 4ac$$

“h” and “k” are the coordinates of the vertex.

The above formula can also be written as follows:


The other method to find the vertex of a parabola is as follows:

We know that the x-coordinate of a vertex, (i.e) \(h\) is \(\frac{−b}{2a}\).

Now, substitute the x-coordinate value in the given standard form of the parabola equation $$y=ax^2+bx+c$$, we will get the y-coordinate of a vertex.

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Solved Examples Using Vertex Formula

\(\mathbf{\color{red}{Find the vertex of a parabola, y=3x^2+12x-12.}}\)

Given parabola equation: $$y=3x^2+12x-12$$

The given parabola equation is of the standard form $$y=ax^2+bx+c$$

By comparing the given equation and standard form, we get

$$a = 3\ b= 12\ c = -12$$

We know that the vertex formula is $$(\frac{-b}{2a}, \frac{-D}{4a})$$

We know that $$D = b^2 – 4ac$$

Therefore, $$D = 12\times 2-(4\times 3\times (-12))$$

$$D = 144+144$$

$$D = 288$$

Now, substitute all the known values in the formula, we get

$$Vertex,\ (h, k) = (\frac{-12}{2\times 3}, \frac{-288}{4\times 3})$$

$$(h, k) = (\frac{-12}{6}, \frac{-288}{12})$$

$$(h. k) = (-2, -24)$$

\(\mathbf{\color{blue}{The\ vertex\ (h, k)\ of\ the\ parabola\ y=3x^2+12x-12\ is\ (-2, -24)}}\).

\(\mathbf{\color{red}{Find\ the\ vertex\ of\ the\ parabola\ y=3x^2-6x+1}}\).

Given parabola equation: $$y = 3x^2-6x+1$$

The standard form of a parabola is $$y=ax^2+bx+c$$

By comparing standard form and given parabola equations, we get $$a = 3,\ b=-6,\ c = 1$$

We know that the formula to calculate the x-coordinate of a vertex is \(\frac{-b}{2a}\).

Hence, $$h = \frac{-(-6)}{2 \times 3})

$$h = \frac{6}{6} = 1$$

Therefore, the x-coordinate of a \(vertex\ is\ 1\).

Now, we need to find the y-coordinate of a vertex. (i.e.) \(k\).

To get the value of the y-coordinate, substitute \(x =1\) in the given equation \(y = 3x^2-6x+1\).

Hence, y-coordinate $$(k) = 3\times 1\times 2-(6\times 1)+1)$$

y-coordinate $$(k) = 3-6+1 = -2$$

\(\mathbf{\color{blue}{Hence,\ the\ coordinate\ of\ the\ vertex\ of\ a\ parabola\ (h, k)\ is\ (1, -2)}}\).

Important Points To Remember For Vertex Of Parabola

  • The standard form of a parabola is $$y=ax^2+bx+c$$
  • The vertex form of a parabola is $$y = a(x-h)^2 + k$$
  • The vertex formula is used to find the vertex of a parabola. The formula to find the vertex is $$(\frac{-b}{2a}, \frac{-D}{4a}),\ where\ D = b^2-4ac.$$

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What is a vertex in math?

A vertex (or node) of a graph is one of the objects that are connected together. The connections between the vertices are called edges or links. A graph with 10 vertices (or nodes) and 11 edges (links).

How do you find the vertex?

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 a vertex example?

Vertex typically means a corner or a point where lines meet. For example, a square has four corners, each is called a vertex. The plural form of a vertex is vertices. A square for example has four vertices.

What is a vertex on a graph?

“Vertex” is a synonym for a node of a graph, i.e., one of the points on which the graph is defined and which may be connected by graph edges. The terms “point,” “junction,” and 0-simplex are also used.

What is the difference between a Vertice and an angle?

Vertex is a point, where two adjacent sides of a polygon meet. Angle is a measure of rotation in between those two adjacent sides at that vertex.