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Vertex of a Parabola

Before going to learn what is the vertex of a parabola, let us recall what is a parabola. A parabola is basically a 'U' shaped curve turned in different directions. It can be in one of the 4 forms.

'U' shaped (top opened) parabola
'∩' shaped (bottom opened) parabola
'⊃' shaped (left opened) parabola
'⊂' shaped (right opened) parabola

Every parabola has a turning point. i.e., it has a point where it either changes from "increasing" to "decreasing" or vice versa. That turning point is called the vertex of the parabola. Let us learn more about the vertex of a parabola along with the different processes of finding it.

1.	What is Vertex of a Parabola?
2.	Vertex of a Parabola Formula
3.	Finding Vertex of a Parabola From Standard Form
4.	Finding Vertex of a Parabola From Vertex Form
5.	Finding Vertex of a Parabola From Intercept Form
6.	Properties of Vertex of a Parabola
7.	FAQs on Vertex of a Parabola

What is Vertex of a Parabola?

The vertex of a parabola is the point at which the parabola makes its sharpest turn. A parabolic function has either a maximum value (if it is of the shape '∩') or a minimum value (if it is of the shape 'U"). The vertex of a parabola is also the point of intersection of the parabola and its axis of symmetry.

The vertex of a parabola definition is shown as the intersecting point of the parabola and the axis of symmetry.

Every parabola has exactly one vertex.

Different Types of Parabolas

There can be two types of equations of a parabola which represent 4 different types of parabolas. The equation of any parabola involves a quadratic polynomial.

Top/Bottom opened parabolas are of the form y = ax² + bx + c
Left/right opened parabolas are of the form x = ay² + by + c

Top/Bottom Opened Parabolas:

The equation of a top/bottom opened parabola can be in one of the following three forms:

Standard form: y = ax² + bx + c
Vertex Form: y = a (x - h)² + k
Intercept Form: y = a (x - p)(x - q)

In each of the cases, the parabola opens up if a > 0, and it opens down if a < 0. These types of parabolas are quadratic functions.

Left/Right Opened Parabolas:

The equation of a left/right opened parabola can be in one of the following three forms:

Standard form: x = ay² + by + c
Vertex Form: x = a (y - k)² + h
Intercept Form: x = a (y - p)(y - q)

In each of the cases, the parabola opens to the right side if a > 0, and it opens to the left side if a < 0.

☛Note: These parabolas do NOT represent functions as they fail vertical line test.

Vertex of a Parabola Formula

Here are the formulas to find the vertex of any type of parabola when it is in different forms. We are going to learn about each of them in detail in the upcoming sections.

	Top/Bottom Open Parabola	Left/Right Open Parabola
Standard Form	f(x) = ax² + bx + c Vertex = (-b/2a, f(-b/2a))	f(y) = ay² + by + c Vertex = (f(-b/2a), -b/2a)
Vertex Form	f(x) = a(x - h)² + k Vertex = (h, k)	f(y) = a(y - k)² + h Vertex = (h, k)
Intercept Form	f(x) = a (x - p) (x - q) Vertex = \(\left(\frac{p+q}{2}, f\left(\frac{p+q}{2}\right)\right)\)	f(y) = a (y - p) (y - q) Vertex = \(\left(f\left(\frac{p+q}{2}\right), \frac{p+q}{2}\right)\)

Finding Vertex of a Parabola From Standard Form

We know that the equation of a parabola in standard form can be either of the form y = ax² + bx + c (up/down) or of the form x = ay² + by + c (left/right). How to find vertex from standard form? Let's see.

Vertex of a Top/Bottom Opened Parabola

When a parabola opens up or down, its equation in the standard form is of the form y = ax² + bx + c. Here are the steps to find the vertex (h, k) of such parabolas. The steps are explained with an example where we will find the vertex of the parabola y = 2x² - 4x + 1.

Step - 1: Compare the equation of the parabola with the standard form y = ax² + bx + c.
By comparing y = 2x² - 4x + 1 with the above equation, a = 2, b = -4, and c = 1.
Step - 2: Find the x-coordinate of the vertex using the formula, h = -b/2a
Then we get h = -(-4) / (2 × 2) = 1.
Step - 3: To find the y-coordinate (k) of the vertex, substitute x = h in the expression ax²+ bx + c.
Then k = 2(1)² - 4(1) + 1 = 2 - 4 + 1 = -1.
Step - 4: Write the vertex (h, k) as an ordered pair.
The vertex = (h, k) = (1, -1).

Vertex of a Left/RightOpened Parabola

When a parabola opens left or right, its equation in the standard form is of the form x = ay² + by + c. Here are the steps to find the vertex (h, k) of such parabolas which are explained with an example where we will find the vertex of the parabola x = 2y² - 4y + 1.

Step - 1: Compare the equation of the parabola with the standard form x = ay² + by + c.
By comparing x = 2y² - 4y + 1 with the above equation, a = 2, b = -4, and c = 1.
Step - 2: Find the y-coordinate of the vertex using the formula, k = -b/2a
Then we get k = -(-4) / (2 × 2) = 1.
Step - 3: To find the x-coordinate (h) of the vertex, substitute y = k in the expression ay²+ by + c.
Then h = 2(1)² - 4(1) + 1 = 2 - 4 + 1 = -1.
Step - 4: Write the vertex (h, k) as an ordered pair.
The vertex = (h, k) = (-1, 1).

Finding Vertex of a Parabola From Vertex Form

We know that the equation of a parabola in vertex form can be either of the form y = a(x - h)² + k (up/down) or of the form x = a(y - k)² + h (left/right). Let us see the steps to find the vertex of the parabola in each case.

Vertex of a Top/Bottom Opened Parabola

When a parabola opens to the top or bottom, its equation in the vertex form is of the form y = a(x - h)² + k. Here are the steps to find the vertex (h, k) of such parabolas. The steps are explained with an example where we will find the vertex of the parabola y = 2(x + 3)² + 5

Step - 1: Compare the equation of the parabola with the vertex form y = a(x - h)² + k and identify the values of h and k.
By comparing y = 2(x + 3)² + 5 with the above equation, h = -3 and k = 5.
Step - 2: Write the vertex (h, k) as an ordered pair.
The vertex = (h, k) = (-3, 5).

Vertex of a Left/RightOpened Parabola

When a parabola opens to the left or to the right side, its equation in the vertex form is of the form x = a(y - k)² + h. Here are the steps to find the vertex (h, k) of such parabolas. The steps are explained with an example where we will find the vertex of the parabola x = 2(y + 3)² + 5

Step - 1: Compare the equation of the parabola with the vertex form x = a(y - k)² + h and identify the values of h and k.
By comparing x = 2(y + 3)² + 5 with the above equation, h = 5 and k = -3.
Step - 2: Write the vertex (h, k) as an ordered pair.
The vertex = (h, k) = (5, -3).

Finding Vertex of a Parabola From Intercept Form

We know that the equation of a parabola in intercept form can be either of the form y = a (x - p) (x - q) (up/down) or of the form y = a(y - p)(y - q) (left/right). Let us see the steps to find the vertex of the parabola in each case.

Vertex of a Top/Bottom Opened Parabola

When a parabola opens to the top or bottom, its equation in the intercept form is of the form y = a (x - p) (x - q), where (p, 0) and (q, 0) are the x-intercepts of the parabola. Here are the steps to find the vertex (h, k) of such parabolas. The steps are explained with an example where we will find the vertex of the parabola y = -(x + 3) (x - 7)

Step - 1: Compare the equation of the parabola with the intercept form y = a (x - p) (x - q) and identify the values of p and q.
By comparing y = -(x + 3) (x - 7) with the above equation, p = -3 and q = 7.
Step - 2: Find the x-coordinate of the vertex, h using the formula h = (p + q)/2.
Then h = (-3 + 7)/2 = 4/2 = 2.
Step - 3: Find the y-coordinate of the vertex, k by substituting x = h in the expression a (x - p) (x - q).
Then k = -(2 + 3) (2 - 7) = 25.
Step - 4: Write the vertex (h, k) as an ordered pair.
The vertex = (h, k) = (2, 25).

Vertex of a Left/Right Opened Parabola

When a parabola opens to the left or right side, its equation in the intercept form is of the form x = a (y - p) (y - q), where (0, p) and (0, q) are the y-intercepts of the parabola. Here are the steps to find the vertex (h, k) of such parabolas. The steps are explained with an example where we will find the vertex of the parabola x = -(y + 3) (y - 7)

Step - 1: Compare the equation of the parabola with the intercept form x = a (y - p) (y - q) and identify the values of p and q.
By comparing x = -(y + 3) (y - 7) with the above equation, p = -3 and q = 7.
Step - 2: Find the y-coordinate of the vertex, k using the formula k = (p + q)/2.
Then k = (-3 + 7)/2 = 4/2 = 2.
Step - 3: Find the x-coordinate of the vertex, h by substituting y = k in the expression a (y - p) (y - q).
Then h = -(2 + 3) (2 - 7) = 25.
Step - 4: Write the vertex (h, k) as an ordered pair.
The vertex = (h, k) = (25, 2).

Properties of Vertex of a Parabola

Here are some properties of the vertex of a parabola that follow from the definition of the vertex of a parabola.

The vertex of a parabola is its turning point.
Since the vertex of a parabola is its turning point, the derivative of the function representing the parabola at the vertex is 0.
A top/bottom open parabola either has a maximum or a minimum at its vertex.
The vertex of a left or right open parabola is neither a maximum nor a minimum to it.
Any type of parabola intersects its axis of symmetry at its vertex.

Important Notes on Vertex of a Parabola:

The vertex of a parabola f(x) = ax² + bx + c is (-b/2a, f(-b/2a)). Its axis of symmetry is x = -b/2a.
Instead of using the formula x = -b/2a, we can convert the standard form f(x) = ax² + bx + c into vertex form f(x) = a (x - h)²+ k by completing the square to find the vertex (h, k).
The vertex of a parabola f(y) = ay² + by + c is (f(-b/2a), -b/2a). Its axis of symmetry is y = -b/2a.
The vertex of a parabolic function f(x) = a (x - h)² + k is (h, k), where 'h' represents the horizontal shift and 'k' is the vertical shift of the parent function f(x) = x².
A top/open parabola y = a(x - h)² + k has
- maximum value at the vertex (h, k) when a < 0 and
- minimum value at the vertex (h, k) when a > 0.
A left/right open parabola has neither maximum nor minimum.
We can use the vertex of a parabola to graph it. For this, form a table of two columns labelled x and y with at least 5 rows. In the x-column, one of the numbers should be the x-coordinate of the vertex and two random numbers on each side (left and right) of it. Find the y-coordinate of each of the above five x-values by substituting each of them into the equation. Plot all the points and join them by a curve.

☛Related Topics:

Examples on Vertex of a Parabola

Example 1: Does the vertex of each of the following parabolas is a maximum/minimum? a) y = 3x² - 4x + 5 b) y = (-1/2)x² + 7

Solution:

We know that y = ax² + bx + c

a) By comparing y = 3x² - 4x + 5 with y = ax² + bx + c, we get a = 3.

Here a = 3 > 0 and hence it has a minimum at its vertex.

b) By comparing y = (-1/2)x² + 7 with y = ax² + bx + c, we get a = (-1/2).

Here a = (-1/2) < 0 and hence it has a maximum at its vertex.

Answer: (a) minimum (b) maximum
Example 2: Find the vertex of the parabola y = 0.5x² + 3x + 4.

Solution:

Comparing the given equation with y = ax² + bx + c, we have a = 0.5, b = 3, and c = 4.

The x-coordinate of the vertex is, h = -b/2a = -3/2(0.5) = -3/1 = -3.

The y-coordinate of the vertex is, k = 0.5(-3)² + 3(-3) + 4 = -0.5.

The vertex of the given parabola is, (h, k) = (-3, -0.5)

Answer: ∴ The vertex of parabola = (-3, -0.5).
Example 3: What is the vertex of the parabola x = 3(y - 2) (y + 4)?

Solution:

Comparing the equation with x = a (y - p) (y - q), we get p = 2 and q = -4.

Then the y-coordinate of the vertex, k = (p + q)/2 = (2 + (-4))/2 = -1.

The x-coordinate of the vertex is, h = 3 (-1 - 2) (-1 + 4) = 3(-3)(3) = -27

The vertex of the given parabola is, (h, k) = (-27, -1)

Answer: ∴ The vertex of parabola = (-27, -1).

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FAQs on Vertex of a Parabola

How to Find the Vertex of a Parabola?

To find the vertex of a parabola that is in standard form y = ax² + bx + c:

Use h = -b/2a for finding h
Substitute x = h in the given equation to find k.
Then (h, k) is the vertex.

Define Vertex of a Parabola.

The vertex of a parabola is its sharp turning point. It is the point where the parabola intersects its axis of symmetry.

What is Vertex of a Parabola Formula?

The vertex of a parabola y = f(x) = ax² + bx + c is (h, k), where h = -b/2a and k = f(-b/2a).

How to Find the Vertex of a Parabola From Vertex Form?

To find the vertex of a parabola that is in vertex form y = a (x - h)² + k:

Compare the given equation with y = a (x - h)² + k and identify the values of h and k.
(h, k) is the vertex.

What are the Properties of Vertex of a Parabola?

Some properties of the vertex of a parabola is:

An up/down parabola has a max/min at its vertex.
The vertex is the turning point of the parabola.
The parabola intersects its axis of symmetry at its vertex.

How to Find the Vertex of a Parabola From Intercept Form?

To find the vertex (h, k) of a parabola that is in intercept form y = a(x - p) (x - q):

Use h = (p + q) / 2 for finding h
Substitute x = h in the equation of parabola to find k.

How to Graph a Parabola Using its Vertex?

To graph a parabola y = a(x - h)² + k using its vertex:

Write a table with two columns labelled x and y.
Write "h" as one of the numbers in the column labelled x.
Write two random numbers less than 'h' and two random numbers greater than 'h' in the same column labelled x.
Fill in the column labelled y by substituting each of the numbers for x in the given equation.
We now have 5 points altogether along with the vertex, plot them all on a graph sheet and connect them by a curve which leads to a parabola.

How to Find the Focus of a Parabola Using its Vertex?

Let (h, k) be the vertex of a parabola. Then

The focus of the parabola y = a (x - h)² + k is given by (h, k + (1/4a))
The focus of the parabola x = a (y - k)² + h is given by (h + (1/4a), k)

How to Find the Axis of Symmetry of a Parabola Using its Vertex?

Here are the formulas to find the axis of symmetry of a parabola using its vertex:

The axis of symmetry of an up/down open parabola with vertex (h, k) is x = h.
The axis of symmetry of a left/right open parabola with vertex (h, k) is y = k.

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