How to Find the Vertex of a Parabola in Standard Form

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how to find the vertex of a parabola

People who are Mathematics lovers are gonna find this article quite amazing. Don’t you think playing with numbers, shapes in mathematics is fun? I being a mathematics lover can truly understand how exciting the moment becomes when you solved some complex mathematical problems. Mathematics in itself has various categories such as Geometry, Trigonometry, algebra, calculus, logical, number theory etc. Here also the choice of favourite and interesting category differs from person to person. We are going to cover each part of mathematics in our upcoming articles. So Stay Tuned!!

So, We are here to help you to understand mathematics in the easiest way possible. It might also help you to learn mathematics with fun and a bit of understanding. I am sure eventually you end building your interest in mathematics.

Before we begin with The Vertex of Parabola, It would be better if we understand the meaning of Parabola, basic things related to it and related terms. 

What do you understand by Parabola?

Parabola is a mirror-symmetrical, plane curve and typically U-shaped. A parabola is a curve where any point is at an equal distance from

  1. Fixed Point (the focus)
  2. A fixed straight line (the directrix)

 There are two forms of Parabola.

  • Standard Form:  The standard equation of Parabola is: y=ax2+bx+c
  • Vertex Form: The Vertex form of the quadratic equation of Parabola is: y = (x – h)2 + k, here (h,k) are the points on the x-axis and y-axis respectively.

As we have seen Parabola has two different forms of equations. The method to find Vertex is different for both forms of equations. We are going to explain how to find vertex for standard form and vertex form.

Standard Form:  y = ax2 + bx + c & x = uy2 + vy + w for y-axis and x-axis respectively. You might wonder how to find the Vertex of Parabola? or  how to find the vertex of a parabola in standard form? Here is the answer.

Here The vertex of a parabola is (-b/2a, -v/2u). -b/2a is x-coordinate and -v/2u is y-coordinate.

Vertex form: y = (x – h)2 + k. You might have heard your friend asking you how to find the vertex of a parabola from a given equation. Now, you can help your friend because below we are going to tell you the easiest way to find the vertex of a parabola.

Here The vertex of a parabola is (h,k).

In mathematics, things become more clear if we explain something or try to apply some formula in some real equation. Here, we are going to do the same now. 

For example, If the equation is y = 2(x – 1)2 + 5, the value of his 1, and k is 5. Hence, The Vertex of a Parabola = (1,5)

And if the equation is y = 2x2 – 4x + 7, here first you need to apply to complete the square method to write this given equation into vertex form. 

Lets begin

y – 7 = 2x2 – 4x

y – 7 = 2(x2 – 2x), {Take half of the coefficient of the x-term inside the parentheses, square it and add both side}

y – 7 + 

2 = 2(x2 – 2x) + 2

Y – 5 = 2( x2 – 2x + 1)

Y – 5 = 2( x – 1)2

Y = 2( x – 1)2 + 5

Here The Vertex of a Parabola  is (1, 5)   

While solving geometry, you might encounter problems such as how to find the equation of a parabola given the vertex? Well, it is quite easy. As we know, the Parabola equation and vertex (h,k) are given to us. We just have to put the values of h & k in the parabola equation. Or in simple terms Substitute the vertex’s coordinates for h and k in the vertex form.

For example, let the given vertex be (4, 5). Substituting 4 for h and 5 for k into y = a(x – h)2 + k the equation comes out to be y = a(x – 4)2 + 5.

Let us also learn a few more important terms related to Parabola. Such as:

Axis of Symmetry

Focus

Directrix

Vertex

Latus Rectum

Focal Length

Let’s start understanding what each term means. 

Axis Of Symmetry: The axis of symmetry of a Parabola is a vertical line which divides Parabola into two congruent (equal shape and equal size) halves. Since we are going to discuss upward and downward Parabola. Hence, the axis of symmetry is going to be a vertical line. It can be either x=4a or x=-4a. For the horizontal axis of symmetry, the equation will be y = 4b or y = -4b.

Focus: Focus is a fixed point in Parabola. The Parabola is a set of all points in a plane which are equidistant from a given point and a given line. That given point is termed as “Focus” of the Parabola. It lies on the axis of symmetry. 

Directrix: The Parabola is a set of all points in a plane which are equidistant from a given point and a given line. That given line is termed as “Directrix ” of the Parabola. It does not touch the parabola. Directrix is always perpendicular to the axis of symmetry of a Parabola.

Vertex:  The Vertex of Parabola is the point from where Parabola crosses its axis of symmetry. It is the highest or lowest point on Parabola, also called Maximum or Minimum of a Parabola.

Latus Rectum: 

The Latus rectum of a parabola is a line segment which is perpendicular to the axis of the parabola, passes through the focus and whose endpoints lie on the parabola. For Parabola equation y2 = 4ax or x2 = 4ay, The length of Latus Rectum is 4a.

Focus Length: The distance between the Vertex and the Focus, which is measured along the axis of symmetry, is termed as the “Focus Length” of a parabola.

It is very easy to locate the Focus point of Parabola when the Parabola equation is given. The origin or fixed point of Parabola can be found by pairing the h value with the k value, to give the coordinate (h, k). There are chances of mistakes which can arise from taking the wrong sign of the ‘h.’ For example equation, y = (x – 7)2 + 8, we noticed that the ‘h’ is 7, but it is often mistaken that the x-coordinate of the vertex is -7. This is because our standard form for the equation is y = (x – h)2 + k.

How to Find the Maximum or Minimum?

They-coordinate of the vertex tells us how high or how low the parabola is.

For example, y = (x-7)2 + 8, this equation implies that the y-coordinate of the vertex dictates how high or low on the coordinate plane that the parabola sits. This parabola is resting on the line y = 8. This we calculated by taking x=h. Once we have identified what they-coordinate is, then we have to see whether this number represents a maximum or minimum. This number is termed as a maximum if the parabola is facing downward (the vertex represents the highest point on the parabola), and we call it a minimum if the parabola is facing upward (the vertex represents the lowest point on the parabola).

How do you tell if the parabola is pointed upward or downward by just looking at the equation?

The first thing we need to do is to represent our equation in y = (x – h)2 + k this form. Any given equation can be converted into the same form by following completing the square steps. Now, we need to check if there is a negative sign in front of the parenthetical term. If the equation turns out to be in the form of y = – (x – h)2 + k, the negative in front of the parenthesis tells us that the parabola is pointed downward. If there is no negative sign in front, then the parabola faces upward.

I hope you find the article “How to Find the Vertex of a Parabola in Standard Form” helpful. We tried to cover maximum basic things which one should know about parabola. We are trying our best to provide you quality content, so that things become easier for you.You can share your suggestion with us anytime. Keep Reading!! Keep Learning!!