The vertex of a parabola is the highest point or the lowest point, also known as the maximum or minimum of the parabola. The vertex is the point of intersection of the parabola and its line of symmetry. The vertex can be found in different ways depending on whether the parabola is written in standard form or in vertex form.

Here, we will learn about some important properties of vertices. Then, we will learn how to find the vertices using two methods. Finally, we will apply these methods to solve some problems.

## Properties of the vertex of a parabola

**•** The vertex is the maximum or minimum point of a parabola.

**•** The vertex is the point where the parabola changes direction.

**•** The axis of symmetry intersects the vertex.

## How to find the vertex of a parabola?

The vertex of a parabola can be found using the equation of the parabola. The formulas used are different depending on whether the equation is written in standard form or in vertex form.

### Finding the vertex using standard form

If we have a parabola written in standard form $latex y = a{{x}^2}+bx+c$, we can find the *x*-coordinate of the vertex using the formula $latex x = – \frac{b}{2a}$. Then, we find the value of *y* by substituting the *x* value of the vertex into the standard form.

### Finding the vertex using the vertex form

The vertex form of a parabola allows us to find the vertex easily. If we have the equation $latex y = a {{(x-h)}^2} -k$, the vertex is $latex (h, ~k)$.

## Vertex of parabolas – Examples with answers

The following examples are used to apply the methods used to find the vertex of a parabola. Each example has its respective solution, but it is recommended that you try to solve the problems yourself before looking at the answer.

**EXAMPLE 1**

What is the vertex of the parabola $latex y = 3{{(x-3)}^2}+5$?

##### Solution

This parabola is written in the vertex form $latex y = a{{(x-h)}^2}+k$. In this form, we know that the vertex is $latex (h, k)$. Comparing with this equation, we have the values:

$latex h=3$

$latex k=5$

The vertex is (3, 5).

**EXAMPLE 2**

A parabola is defined by $latex y=4{{(x+4)}^2} -6$. What is its vertex?

##### Solution

Again, we compare the given equation with the vertex form $latex y = a {{(xh)}^2} + k$ and obtain the values of *h* and of *k:*

$latex h=-4$

$latex k=-6$

The vertex is (-4, -6).

**EXAMPLE 3**

What is the vertex of the parabola $latex y = 2{{x}^2}+4x+5$?

##### Solution

This parabola is written in standard form. We can obtain the *x* coordinate of the vertex using the formula $latex x = – \frac{b}{2a}$. Therefore, we have:

$latex x=-\frac{b}{2a}$

$latex =-\frac{4}{2(2)}$

$latex =-\frac{4}{4}$

$latex =-1$

Now, we substitute the value of *x* into the equation to find the coordinate in *y*:

$latex y=2{{x}^2}+4x+5$

$latex =2{{(-1)}^2}+4(-1)+5$

$latex =2-4+5$

$latex =3$

The vertex is (-1, 3).

**EXAMPLE 4**

If we have the parabola $latex y = -2{{x}^2}+12x-7$, what is its vertex?

##### Solution

Using the formula $latex x = – \frac{b}{2a}$, we can find the *x* coordinate of the vertex. Therefore, we have:

$latex x=-\frac{b}{2a}$

$latex =-\frac{12}{2(-2)}$

$latex =-\frac{12}{-4}$

$latex =3$

We use this value of *x* in the equation to find the coordinate in *y*:

$latex y=-2{{x}^2}+12x-7$

$latex =-2{{(3)}^2}+12(3)-7$

$latex =-18+36-7$

$latex =9$

The vertex is (3, 9).

## Vertex of a parabola – Practice problems

Practice using the methods to find the vertex of parabolas by solving the following problems. If you need help with this, you can look at the solved examples above.

## See also

Interested in learning more about parabolas? Take a look at these pages:

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