Graph & Equation
Graphing Equation Lessons

# Vertex of a Quadratic Equation

### Practice

Now , let's try some MCQ questions to understand this lesson better.
You can start by going through the series of the graphing linear equations questions below

Question 1 on finding the vertex of a quadratic equation

• MATH TOPICS
• Graph & Equations
• Algebra
• Exponents
• Symmetry
• Fractions
• Measurements
• Interest
• Angles
• Triangles

### Lesson Objective

In this lesson, we will learn how to find the vertex of a quadratic equation.

In this lesson, we will:

• Learn what is the vertex of a quadratic equation.
• Learn the formula to find the x-coordinate of the vertex.
• See some examples on using the formula to find the coordinates of the vertex.

### What is in this video ?

• 00:00:04.130
In this lesson, we will learn about the vertex of a quadratic equation
• 00:00:09.240
Now, this is the graph of the quadratic equation, y= -x^2 +2x.
• 00:00:17.160
Notice that, this is the highest point on the graph. Hence, this point is the vertex of a quadratic equation..
• 00:00:27.000
Also, if you have a quadratic graph as shown here, the lowest point, is also the vertex of the quadratic equation, y = 2x^2 -8x +3.
• 00:00:38.160
Now, there is a formula to calculate the x-coordinate of the vertex of a quadratic equation.
• 00:00:44.090
The formula is, x = -b/2a. Now, what are 'a' and 'b'? Let's find out.
• 00:00:54.010
The general equation of a quadratic equation is given as, y = ax^2 +bx, +c. Where, 'a', 'b' and 'c' are the coefficients for each term respectively.
• 00:01:10.140
Let's see an example on using this formula, by using this equation, y = 2x^2 -8x +3.
• 00:01:20.180
To find out the values of 'a', 'b' and 'c', we can rewrite this equation as, y = 2x^2 + (-8)x +3.
• 00:01:32.200
By comparing this equation with the general equation, we can see that, 'a' is equals to 2, 'b' is equals to -8, and 'c' is equals to 3.
• 00:01:44.200
Knowing this, we can substitute 'b', with -8, and substitute 'a', with 2. Now, we can simplify this term.ves you 3. Let's put this new value y = 3 into the table.
• 00:01:57.000
This term is the same as, [-(-8)]/[2(2)]. Negative multiply bracket -8, gives 8. 2 multiply with 2, gives 4. 8 divide by 4, gives 2
• 00:02:17.090
Finally, we have x equals to 2. With this, we know that the vertex of the quadratic equation has the x-coordinate of 2..
• 00:02:27.220
Let's go back to the original equation. Now, we can use the x–coordinate to find the y-coordinate of the vertex.
• 00:02:38.060
To do so, we can substitute x with 2. -8 multiply with 2, gives -16. Adding -16 with 3, gives -13.
• 00:02:53.150
Again, we can substitute x with 2. 2^2 is the same as, 2 multiply by 2. Which is equals to 4. Let’s write down this number.
• 00:03:08.100
Let's continue. 2 multiply with 4, gives 8. 8 minus 13, gives -5.
• 00:03:20.170
Hence, the vertex has y-coordinate of -5. This y-coordinate is located here.
• 00:03:30.080
Hence, the vertex of the equation has the coordinates of (2, -5).
• 00:03:40.080
Let's see another example. Consider the quadratic equation, y = -x^2 +2x. The graph for this equation is shown here.
• 00:03:54.170
Now, the vertex is located at the highest point on the graph. Let's find the coordinates of this vertex
• 00:03:54.170
Now, the vertex is located at the highest point on the graph. Let's find the coordinates of this vertex
• 00:04:03.210
Again, we start with the formula for the x-coordinate of the vertex of a quadratic equation, x = -b/2a.
• 00:04:13.050
To find the values of 'a' and 'b', we can compare this equation with the general equation, y = ax^2 + bx +c.
• 00:04:24.230
For easier comparison, we can rewrite this quadratic equation as, y = -1x^2 +4x + 0. Hence, we can see that, 'a' is equals to -1, ‘b’ is equals to 2, and 'c' is equals to 0.
• 00:04:44.120
Knowing this, we can substitute 'b' with 2 and substitute 'a' with -1.
• 00:04:53.170
Let's simplify this term. Now, this term is the same as, -2/2(-1). 2 multiply by -1, gives -2. -2 divides by -2, gives 1. Hence, we have, x equals to 1.
• 00:05:19.130
With this, the vertex has the x-coordinate of 1. Let's change back to the original equation.
• 00:05:27.130
Now, we can use this x-coordinate to find the y-coordinate of the vertex. Here’s how. Substitute x with 1. 2 multiply with 1, gives back 2.
• 00:05:43.240
Let's continue. Again, substitute x with 1. 1^2, is the same as, 1 multiply with 1. This gives 1. Let’s write this down.
• 00:06:01.170
Finally, -1 add with 2, gives 1.
• 00:06:07.000
Hence, the y-coordinate of the vertex is 1. This can be seen here.
• 00:06:15.070
Hence, the coordinates of the vertex of this quadratic equation is (1, 1).
• 00:06:24.010
That is all for this lesson. Try out the practice question to test your understanding.

## Study Tips

The general quadratic equation is given as:

Where a, b and c are the coefficients of each term respectively. Now, the graph of a quadratic equation will always have a highest point or a lowest point depending on the value of a. This point is called the Vertex. The pictures below will illustrate this.

Graph with a Highest Point

Graph with a Lowest Point

## Vertex Formula

The formula to find the x-coordinate of the vertex is shown below:

Where a and b are the coefficients in the general quadratic equation shown below:

Watch the video to find out where these plane are located.
Visit the Q&A featuring categories such as Algebra, Graphs, Exponents and more. Also you can submit math question, share or give comments there.
Practice

Now , let's try some MCQ questions to understand this lesson better.
You can start by going through the series of the vertex of a quadratic equation question below

Question 1 on finding the vertex of a quadratic equation