Quadratic Function: Vertex Form
How to find the vertex of a quadratic function (by changing it to vertex form): formula, 3 examples, and their solutions.
Formula
y = a(x - h)2 + k
is a quadratic function in vertex form.
The vertex of the quadratic function is
(h, k).
Exampley = x2 - 4x + 5
To find the vertex of the quadratic function,
change the quadratic function to vertex form.
Use x2 - 4x
to make a perfect square trinomial.
x2 is x2.
-4x is
-2 times
x times,
(-4x)/(-2⋅x), 2.
Quadratic Equation: Completing the Square
Write +22.
To undo +22,
write -22.
Write +5.
So y = x2 - 4x + 5 becomes
y = x2 - 2⋅x⋅2 + 22 - 22 + 5.
x2 - 2⋅x⋅2 + 22
= (x - 2)2
Factor a Perfect Square Trinomial
-22 = -4
-4 + 5 = +1
y = (x - 2)2 + 1
Then the vertex of the quadratic function is
(2, 1).
So (2, 1) is the answer.
Exampley = x2 + 6x - 1
To find the vertex,
change the quadratic function to vertex form.
Use x2 + 6x
to make a perfect square trinomial.
x2 is x2.
+6x is
+2 times
x times,
(+6x)/(+2⋅x), 3.
Write +32.
To undo +32,
write -32.
Write -1.
So y = x2 + 6x - 1 becomes
y = x2 + 2⋅x⋅3 + 32 - 32 - 1.
x2 + 2⋅x⋅3 + 32
= (x + 3)2
-32 = -9
To see the x value of the vertex easily,
change (x + 3)2 to (x - (-3))2.
-9 - 1 = -10
y = (x - (-3))2 - 10
Then the vertex of the quadratic function is
(-3, -10).
So (-3, -10) is the answer.
Exampley = -x2 + 8x - 16
To find the vertex,
change the quadratic function to vertex form.
First change -x2 to -(x2.
+8x = -(-8x)
-8x is
-2 times
x times,
(-8x)/(-2⋅x), 4.
To make a perfect square trinomial
in the parentheses,
write +42).
To undo -(+42),
write +42.
Write -16.
So y = -x2 + 8x - 16 becomes
y = -(x2 - 2⋅x⋅4 + 42) + 42 - 16.
-(x2 - 2⋅x⋅4 + 42)
= -(x - 4)2
+42 = +16
+16 - 16 = 0
y = -(x - 4)2
Then the vertex of the quadratic function is
(4, 0).
So (4, 0) is the answer.