Question

y = 3 x ^ 2 -4x + 1find the vertex of the function. convert to vertex form

Answer 1

$y=3x^2-4x+1$

To find the vertex of a parabola (quadratic equiation) you use the next:

1. Find the axis of simetry (value of x in the vertex) with the next formula:

$\begin{gathered} y=ax^2+bx+c \\ \\ x=\frac{-b}{2a} \end{gathered}$

For the given equation:

b=-4

a=3

$x=\frac{-(-4)}{2(3)}=\frac{4}{6}=\frac{2}{3}$

Axis of simetry x=2/3

2. Find the value of y in the vertex. Use the axis of simetry:

$\begin{gathered} y=3(\frac{2}{3})^2-4(\frac{2}{3})+1 \\ \\ y=3(\frac{4}{9})-\frac{8}{3}+1 \\ \\ y=\frac{12}{9}-\frac{8}{3}+1 \\ \\ y=\frac{4}{3}-\frac{8}{3}+\frac{3}{3} \\ \\ y=-\frac{1}{3} \end{gathered}$3. The vertex is (2/3 , -1/3)

4. Write in vertex form.

General vertex form of a quadratic equation:

$y=a(x-h)^2+k$

The vertex is (h,k)

For the given equation:

a=3

h=2/3

k= -1/3

Equation in vertex form:$y=3(x-\frac{2}{3})^2-\frac{1}{3}$