Answer:
[tex]f(x) = 3(x -\frac{2}{3})^2 -\frac{22}{3}[/tex]
The vertex is [tex](\frac{2}{3}, -\frac{22}{3})[/tex]
Step-by-step explanation:
For a general quadratic function the form is:
[tex]ax ^ 2 + bx + c[/tex]
For the function
[tex]f(x) = 3x ^ 2 -4x -6[/tex]
Take common factor 3.
[tex]f(x) = 3(x ^ 2 -\frac{4}{3}x - 2)[/tex]
The values of the coefficients for the function within the parenthesis are the following: [tex]a = 1[/tex], [tex]b = -\frac{4}{3}[/tex], [tex]c = -2[/tex]
Take the value of b and divide it by 2. Then, the result obtained squares it.
[tex]\frac{b}{2}= -\frac{2}{3}[/tex]
[tex](\frac{b}{2})^2=\frac{4}{9}[/tex]
Add and subtract [tex]\frac{4}{9}[/tex]
[tex]f(x) = 3([x ^ 2 -\frac{4}{3}x +\frac{4}{9}]- 2-\frac{4}{9})[/tex]
Write the expression of the form
[tex]f(x) = (x-\frac{b}{2})^2 +k[/tex]
[tex]f(x) = 3(x -\frac{2}{3})^2 -\frac{22}{3}[/tex]
The vertex is [tex](\frac{2}{3}, -\frac{22}{3})[/tex]
