We show you in this lesson two ways to convert a quadratic function from the general form to the vertex form.
Write the function f(x) = 3x2 - 12x + 10 in vertex form by completing the square
f(x) = 3x2 - 12x + 10
First, group the quadratic term and the linear term.
f(x) = (3x2 - 12x) + 10
Since we need a leading coefficient of 1 to complete the square, factor out the 3.
f(x) = 3(x2 - 4x) + 10
Complete the square. Review this lesson about completing the square if needed.
Basically, you need to find the constant term for x2 - 4x.
The constant term is (-4/2)2 = (-2)2
Add (-2)2 to 3(x2 - 4x) + 10. The way we are going to do that is by adding (-2)2 and then removing (-2)2 so we do not change the function.
f(x) = 3(x2 - 4x + (-2)2 - (-2)2) + 10
f(x) = 3(x2 - 4x + (-2)2 - 4) + 10
f(x) = 3(x2 - 4x + (-2)2)+ 3(- 4) + 10
f(x) = 3(x2 - 4x + (-2)2) + -12 + 10
f(x) = 3(x - 2)2 + -2
The vertex form is f(x) = 3(x - 2)2 + -2
Write the function f(x) = 3x2 - 12x + 10 in vertex form using the formula to find the x-coordinate of the vertex.
f(x) = 3x2 - 12x + 10
Find the x-coordinate of the vertex using x = -b/2a
a = 3, b = -12 and c = 10
x = - -12/2(3) = 12/6 = 2
Find the y-coordinate of the vertex using f(x) = 3x2 - 12x + 10
f(2) = 3(2)2 - 12(2) + 10
f(2) = 3(4) - 24 + 10
f(2) = 12 - 24 + 10
f(2) = -12 + 10
f(2) = -2
The vertex form of a quadratic function is f(x) = a(x - h)2 + k, where a, h, and k are real numbers and (h,k) is the vertex.
Therefore, a = 3, h = 2 and k = -2
f(x) = 3(x - 2)2 + -2