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) = 3x^{2} - 12x + 10 in vertex form by completing the square
f(x) = 3x^{2} - 12x + 10
First, group the quadratic term and the linear term.
f(x) = (3x^{2} - 12x) + 10
Since we need a leading coefficient of 1 to complete the square, factor out the 3.
f(x) = 3(x^{2} - 4x) + 10
Complete the square. Review this lesson about completing the square if needed.
Basically, you need to find the constant term for x^{2} - 4x.
The constant term is (-4/2)^{2} = (-2)^{2}
Add (-2)^{2 }to 3(x^{2} - 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(x^{2} - 4x + (-2)^{2} - (-2)^{2}) + 10
f(x) = 3(x^{2} - 4x + (-2)^{2} - 4) + 10
f(x) = 3(x^{2} - 4x + (-2)^{2})+ 3(- 4) + 10
f(x) = 3(x^{2} - 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) = 3x^{2} - 12x + 10 in vertex form using the formula to find the x-coordinate of the vertex.
f(x) = 3x^{2} - 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) = 3x^{2} - 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
Sep 17, 23 09:46 AM
Jun 09, 23 12:04 PM