Vertex of a Parabola Formula

The point where the parabola and its axis of symmetry intersect is called the vertex of a parabola. It is used to determine the coordinates of the point on the parabola’s axis of symmetry where it crosses it. For standard equation of a parabola y = ax² + bx + c, the vertex point is the coordinate (h, k). If the coefficient of x² in the equation is positive (a > 0), then vertex lies at the bottom else it lies on the upper side.

Properties of Vertex of a Parabola

1. The vertex of every parabola is its turning point.

2. The derivative of the parabola function at its vertex is always zero.

3. A parabola which is either open at its top or bottom has a maxima or a minima at its vertex.

4. The vertex of a left or right open parabola is neither a maxima nor a minima of the parabola.

5. Vertex is the point of intersection between the parabola and its axis of symmetry.

Vertex formula

For the vertex form of the parabola, y = a(x – h)² + k, the coordinates (h, k) of the vertex are,

(h, k) = (-b/2a, -D/4a)

where,

a is the coefficient of x²,

b is the coefficient of x,

D = b² – 4ac is the discriminant of the standard form y = ax² + bx + c.

Derivation

Suppose we have a parabola with standard equation as, y = ax² + bx + c.

This can be written as,

y – c = ax² + bx

y – c = a (x² + bx/a)

Adding and subtracting b²/4a² on the RHS, we get

y – c = a (x² + bx/a + b²/4a² – b²/4a²)

y – c = a ((x + b/2a)² – b²/4a²)

y – c = a (x + b/2a)² – b²/4a

y = a (x + b/2a)² – b²/4a + c

y = a (x + b/2a)² – (b²/4a – c)

y = a (x + b/2a)² – (b² – 4ac)/4a

We know, D = b² – 4ac, so the equation becomes,

y = a (x + b/2a)² – D/4a

Comparing the above equation with the vertex form y = a(x – h)² + k, we get

h = -b/2a and k = -D/4a

This derives the formula for coordinates of the vertex of a parabola.

Sample Problems

Problem 1. Find the coordinates of the vertex for the parabola y = 2x² + 4x – 4.

Solution:

We have the equation as, y = 2x² + 4x – 4.

Here, a = 2, b = 4 and c = -4.

Now, it is known that the coordinates of the vertex are given by, (-b/2a, -D/4a) where D = b² – 4ac.

D = (4)² – 4 (2) (-4)

= 16 + 32

= 48

So, x – coordinate of vertex = -4/2(2) = -4/4 = -1.

y – coordinate of vertex = -48/4(2) = -48/8 = -6

Hence, the vertex of the parabola is (-1, -6).

Problem 2. Find the coordinates of the vertex for the parabola y = 3x² + 5x – 2.

Solution:

We have the equation as, y = 3x² + 5x – 2.

Here, a = 3, b = 5 and c = -2.

Now, it is known that the coordinates of the vertex are given by, (-b/2a, -D/4a) where D = b² – 4ac.

D = (5)² – 4 (3) (-2)

= 25 + 24

= 49

So, x – coordinate of vertex = -5/2(3) = -5/6

y – coordinate of vertex = -49/4(3) = -49/12

Hence, the vertex of the parabola is (-5/6, -49/12).

Problem 3. Find the coordinates of the vertex for the parabola y = 3x² – 6x + 1.

Solution:

We have the equation as, y = 3x² – 6x + 1.

Here, a = 3, b = -6 and c = 1.

Now, it is known that the coordinates of the vertex are given by, (-b/2a, -D/4a) where D = b² – 4ac.

D = (-6)² – 4 (3) (1)

= 36 – 12

= 24

So, x – coordinate of vertex = 6/2(3) = 6/6 = 1

y – coordinate of vertex = -24/4(3) = -24/12 = -2

Hence, the vertex of the parabola is (1, -2).

Problem 4. Find the coordinates of the vertex for the parabola y = 3x² + 8x – 8.

Solution:

We have the equation as, y = 3x² + 8x – 8.

Here, a = 3, b = 8 and c = -8.

Now, it is known that the coordinates of the vertex are given by, (-b/2a, -D/4a) where D = b² – 4ac.

D = (8)² – 4 (3) (-8)

= 64 + 96

= 160

So, x – coordinate of vertex = -8/2(3) = -8/6 = -4/3

y – coordinate of vertex = -160/4(3) = -160/12 = -40/3

Hence, the vertex of the parabola is (-4/3, -40/3).

Problem 5. Find the coordinates of the vertex for the parabola y = 6x² + 12x + 4.

Solution:

We have the equation as, y = 6x² + 12x + 4.

Here, a = 6, b = 12 and c = 4.

Now, it is known that the coordinates of the vertex are given by, (-b/2a, -D/4a) where D = b² – 4ac.

D = (12)² – 4 (6) (4)

= 144 – 96

= 48

So, x – coordinate of vertex = -12/2(6) = -12/12 = -1

y – coordinate of vertex = -48/4(6) = -48/24 = -2

Hence, the vertex of the parabola is (-1, -2).

Problem 6. Find the coordinates of the vertex for the parabola y = x² + 7x – 5.

Solution:

We have the equation as, y = x² + 7x – 5.

Here, a = 1, b = 7 and c = -5.

Now, it is known that the coordinates of the vertex are given by, (-b/2a, -D/4a) where D = b² – 4ac.

D = (7)² – 4 (1) (-5)

= 49 + 20

= 69

So, x – coordinate of vertex = -7/2(1) = -7/2

y – coordinate of vertex = -69/4(1) = -69/4

Hence, the vertex of the parabola is (-7/2, -69/4).

Problem 7. Find the coordinates of the vertex for the parabola y = 2x² + 10x – 3.

Solution:

We have the equation as, y = x2 + 7x – 5.

Here, a = 1, b = 7 and c = -5.

Now, it is known that the coordinates of the vertex are given by, (-b/2a, -D/4a) where D = b2 – 4ac.

D = (7)2 – 4 (1) (-5)

= 49 + 20

= 69

So, x – coordinate of vertex = -7/2(1) = -7/2

y – coordinate of vertex = -69/4(1) = -69/4

Hence, the vertex of the parabola is (-7/2, -69/4).

FAQs on Vertex of a Parabola

Question 1. What do you mean by the vertex of a parabola?

Solution:

The point where the parabola and its axis of symmetry intersect is called the vertex of a parabola. It is used to determine the coordinates of the point on the parabola’s axis of symmetry where it crosses it.

Question 2. How does one calculate the vertex of a parabola?

Solution:

For standard equation of a parabola y = ax² + bx + c, the vertex point is the coordinate (h, k).

Question 3. Write the properties of the vertex of a parabola.

Solution:

1. The vertex of every parabola is its turning point.

2. The derivative of the parabola function at its vertex is always zero.

3. A parabola which is either open at its top or bottom has a maxima or a minima at its vertex.

4. The vertex of a left or right open parabola is neither a maxima nor a minima of the parabola.

5. Vertex is the point of intersection between the parabola and its axis of symmetry.

Question 4. The vertex form of a parabola is given. How would you find its vertex?

Solution:

For standard equation of a parabola y = ax² + bx + c, the vertex point is the coordinate (h, k).

Question 5. What do you mean by focus of a parabola?

Solution:

A parabola is set of all points in a plane which are an equal distance away from a given point and given line. The point is called the focus of the parabola.

Question 6. How to graph a parabola with its vertex?

Solution:

1. Find the x and y coordinates.

2. Write two numbers smaller and two greater than focus and mark them as x- coordinates.

3. Substitute the value of function for x and find y coordinates.

4.Identify the focus and vertex of the parabola and plot the coordinates on a graph paper.