How to Find the Vertex of a Quadratic Equation

Co-authored by David Jia

Last Updated: October 19, 2024 Fact Checked

The vertex of a quadratic equation or parabola is the highest or lowest point of that equation. It lies on the plane of symmetry of the entire parabola as well; whatever lies on the left of the parabola is a complete mirror image of whatever is on the right. If you want to find the vertex of a quadratic equation, you can either use the vertex formula, or complete the square.

Method 1

Method 1 of 2:

Using the Vertex Formula

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In a quadratic equation, the $x^{2}$ term = a, the $x$ term = b, and the constant term (the term without a variable) = c. Let's say you're working with the following equation: ' $y=x^{2}+9x+18$ . In this example, $a$ = 1, $b$ = 9, and $c$ = 18.^{[1]
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Research source}
2
Use the vertex formula for finding the x-value of the vertex. The vertex is also the equation's axis of symmetry. The formula for finding the x-value of the vertex of a quadratic equation is $x={\frac {-b}{2a}}$ $x={\frac {-b}{2a}}$ . Plug in the relevant values to find x. Substitute the values for a and b. Show your work:
- $x={\frac {-b}{2a}}$
- $x={\frac {-(9)}{(2)(1)}}$
- $x={\frac {-9}{2}}$
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3
Plug the $x$ value into the original equation to get the $y$ value. Now that you know the $x$ $x$ value, just plug it in to the original formula for the $y$ $y$ value. You can think of the formula for finding the vertex of a quadratic function as being $(x,y)=\left[({\frac {-b}{2a}}),f({\frac {-b}{2a}})\right]$ $(x,y)=\left[({\frac {-b}{2a}}),f({\frac {-b}{2a}})\right]$ . This just means that to get the $y$ $y$ value, you have to find the $x$ $x$ value based on the formula and then plug it back into the equation.^{[2]
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Research source} Here's how you do it:
- $y=x^{2}+9x+18$
- $y={\frac {(-9)}{(2)}}^{2}+9{\frac {(-9)}{(2)}}+18$
- $y={\frac {81}{4}}-{\frac {81}{2}}+18$
- $y={\frac {81}{4}}-{\frac {162}{4}}+{\frac {72}{4}}$
- $y={\frac {(81-162+72)}{4}}$
- $y={\frac {-9}{4}}$
Write down the $x$ and $y$ values as an ordered pair. Now that you know that $x={\frac {-9}{2}}$ , and $y={\frac {-9}{4}}$ , just write them down as an ordered pair: $({\frac {-9}{2}},{\frac {-9}{4}})$ .^{[3]
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Research source} The vertex of this quadratic equation is $({\frac {-9}{2}},{\frac {-9}{4}})$ . If you were to draw this parabola on a graph, this point would be the minimum of the parabola, because the $x^{2}$ term is positive.

Method 2

Method 2 of 2:

Completing the Square

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Completing the square is another way to find the vertex of a quadratic equation. For this method, when you get to the end, you'll be able to find your x and y coordinates right away, instead of plugging the x coordinate back in to the original equation. Let's say you're working with the following quadratic equation: $x^{2}+4x+1=0$ .^{[4]
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Divide each term by the coefficient of the $x^{2}$ term. In this case, the coefficient of the $x^{2}$ term is 1, so you can skip this step. Dividing each term by 1 would not change anything. Dividing each term by 0, however, will change everything.^{[5]
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3
Move the constant term to the right side of the equation. The constant term is the term without a coefficient. In this case, it is 1. Move 1 to the other side of the equation by subtracting 1 from both sides.^{[6]
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Research source} Here's how you do it:
- $x^{2}+4x+1=0$
- $x^{2}+4x+1-1=0-1$
- $x^{2}+4x=-1$
4
Complete the square on the left side of the equation. To do this, simply find $({\frac {b}{2}})^{2}$ $({\frac {b}{2}})^{2}$ and add the result to both sides of the equation. Plug in 4 for $b$ $b$ , since $4x$ $4x$ is the b-term of this equation.^{[7]
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Research source}
- $({\frac {4}{2}})^{2}=2^{2}=4$ $({\frac {4}{2}})^{2}=2^{2}=4$ . Now, add 4 to both sides of the equation to get the following:
  - $x^{2}+4x+4=-1+4$
  - $x^{2}+4x+4=3$
Now you will see that $x^{2}+4x+4$ is a perfect square.^{[8]
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Research source} It can be rewritten as $(x+2)^{2}=3$
Use this format to find the $x$ and $y$ coordinates. You can find your $x$ coordinate by simply setting $(x+2)^{2}$ equal to zero.^{[9]
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Research source} So when $(x+2)^{2}=0$ , what would $x$ have to be? The variable $x$ would have to be -2 to balance out the +2, so your $x$ coordinate is -2. Your y-coordinate is simply the constant term on the other side of the equation. So, $y=3$ . You can also do a shortcut and just take the opposite sign of the number in parentheses to get the x-coordinate. So the vertex of the equation $x^{2}+4x+1=(-2,-3)$ .

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Question

What is a shortcut to find the vertex?

David Jia
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David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math.

David Jia

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Expert Answer

Average out the 2 intercepts of the parabola to figure out the x coordinate. Think of it this way—a parabola is symmetrical, U-shaped curve. So, if you have 2 x intercepts on the left and right sides of this parabola, their average will give you the x coordinate of the vertex, which is directly in the middle. Once you've figured out the x coordinate, you can plug it into the regular quadratic formula to get your y coordinate.

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How do I find x and y intercepts of a parabola?

Donagan

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Find the y-intercept by setting x equal to zero and solving the equation for y. Find the x-intercept by setting y equal to zero and solving for x.

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How can I graph 3(x-1)squared +4 on a ti-84 calculator?

Community Answer

Press the "y=" button. Then,type in "3(x+1)^2+4)". Lastly, hit "zoom," then "0" to see the graph.

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