The graph of a quadratic function is always a parabola. The vertex of a parabola is the point where the parabola crosses its axis of symmetry.
The vertex of a parabola is the highest or lowest point which is also known as maximum or minimum value.
If a parabola opens up, the vertex will be the lowest point and if it opens down, the vertex will be the highest point of the parabola.
Quadratic function in standard form :
f(x) = ax2 + bx + c
Quadratic function in in vertex form :
f(x) = a(x - h)2 + k
where (h, k) is the vertex.
When a quadratic function is given in vertex form, we can find the vertex easily by taking the values of 'h' and 'k'.
When a quadratic function is given in standard form, you can use formula given below to find the x-coordinate of the vertex.
x = -b/2a
After having found the x-coordinate, you can substitute it into the quadratic function and find the y-coordinate of the vertex.
In each case, find vertex of the quadratic function :
Example 1 :
f(x) = 7x2 - 12
Solution :
Comparing f(x) = ax2 + bx + c and f(x) = 7x2 - 12,
a = 1, b = 0 and c = -12
x-coordinate of the vertex :
x = -b/2a
Substitute a = 7 and b = 0.
x = -0/2(7)
x = 0
y-coordinate of the vertex :
Substitute x = 0 in f(x) = 7x2 - 12.
f(0) = 7(0) - 12
= 0 - 12
= -12
y = -12
Vertex of the parabola is (0, -12)
Example 2 :
f(x) = -9x2 - 5
Solution :
Comparing f(x) = ax2 + bx + c and f(x) = -9x2 - 5,
a = -9, b = 0 and c = -5
x-coordinate of the vertex :
x = -b/2a
Substitute a = -9 and b = 0.
x = -0/2(-9)
x = 0
y-coordinate of the vertex :
Substitute x = 0 in f(x) = -9x2 - 5.
f(0) = -9(0) - 5
= 0 - 5
= -5
y = -5
Vertex of the parabola is (0, -5)
Example 3 :
f(x) = (x - 2)2 - 3
Solution :
The given quadratic function is in standard form.
Comparing f(x) = a(x - h)2 + k and f(x) = (x - 2)2 - 3,
h = 2 and k = -3
Vertex of the parabola :
(h, k) = (2, -3)
Example 4 :
f(x) = (x + 3)2 + 4
Solution :
The given quadratic function is in standard form.
Comparing f(x) = a(x - h)2 + k and f(x) = (x + 3)2 + 4,
h = -3 and k = 4
Vertex of the parabola :
(h, k) = (-3, 4)
Example 5 :
f(x) = (2x - 5)2 + 6
Solution :
Write the given quadratic function in vertex form.
f(x) = (2x - 5)2 + 6
f(x) = [2(x - 5/2)]2 + 6
f(x) = [22(x - 5/2)2] + 6
f(x) = 4(x - 5/2)2 + 6
Now, the given quadratic function is in standard form.
Comparing f(x) = a(x - h)2 + k and f(x) = 4(x - 5/2)2 + 6,
h = 5/2 and k = 6
Vertex of the parabola :
(h, k) = (5/2, 6)
Example 6 :
f(x) = (7x + 3)2 + 5
Solution :
Write the given quadratic function in vertex form.
f(x) = (7x + 3)2 + 5
f(x) = [7(x + 3/7)]2 + 5
f(x) = [72(x + 3/7)2] + 5
f(x) = 49(x + 3/7)2 + 5
Comparing f(x) = a(x - h)2 + k and f(x) = 49(x + 3/7)2 + 5,
h = -3/7 and k = 5
Vertex of the parabola :
(h, k) = (-3/7, 5)
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