Sample Problems on Vertex of a Parabola Formula
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).
Vertex of a Parabola Formula
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 the 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 the vertex lies at the bottom else it lies on the upper side.
In this article, we will discuss the vertex of a parabola, its formula, derivation of the formula, and solved examples on it.
Table of Content
- Properties of Vertex of a Parabola
- Vertex of a Parabola Formula
- Derivation of Vertex of a Parabola Formula
- Sample Problems on Vertex of a Parabola Formula