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.
PRECALCULUS
Relevant for…
Learning to find the vertex of a parabola with examples.
See examples
Contents
- Properties of the vertex of a parabola
- How to find the vertex of a parabola?
- Vertex of parabolas – Examples with answers
- Vertex of a parabola – Practice problems
- See also
PRECALCULUS
Relevant for…
Learning to find the vertex of a parabola with examples.
See examples
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 thex-coordinate of the vertex using the formula $latex x = – \frac{b}{2a}$. Then, we find the value ofyby substituting thexvalue 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 ofhand ofk:
$latex h=-4$
$latex k=-6$
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 thexcoordinate 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 ofxinto the equation to find the coordinate iny:
$latex y=2{{x}^2}+4x+5$
$latex =2{{(-1)}^2}+4(-1)+5$
$latex =2-4+5$
$latex =3$
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 thexcoordinate 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 ofxin the equation to find the coordinate iny:
$latex y=-2{{x}^2}+12x-7$
$latex =-2{{(3)}^2}+12(3)-7$
$latex =-18+36-7$
$latex =9$
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.
If we have the parabola $latex y = {{(x-7)}^2}+4$, what is its vertex?
Chooose an answer
If we have the parabola $latex y = {{(x + 4)}^2}-2$, what is its vertex?
Choose an answer
What is the vertex of the parabola $latex y={{x}^2}-2x+3$?
Choose an answer
See also
Interested in learning more about parabolas? Take a look at these pages:
- Parts of a Parabola with Diagrams
- Focus and Directrix of a Parabola
- Characteristics of a Parabola
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
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