To convert the quadratic function [tex]\( f(x) = 3x^2 - 4x + 9 \)[/tex] into its vertex form [tex]\( a(x-h)^2 + k \)[/tex], we need to complete the square. Here's the detailed, step-by-step process:
1. Identify constants in the quadratic function:
- The quadratic function is [tex]\( f(x) = 3x^2 - 4x + 9 \)[/tex].
2. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the first two terms:
- The coefficient of [tex]\( x^2 \)[/tex] is 3.
- Factor 3 out from the first two terms:
[tex]\[ f(x) = 3 (x^2 - \frac{4}{3}x) + 9. \][/tex]
3. Complete the square inside the parentheses:
- To complete the square, take the coefficient of [tex]\( x \)[/tex], which is [tex]\( -\frac{4}{3} \)[/tex], divide it by 2, and square it:
[tex]\[ \left( -\frac{4}{3} \right) \div 2 = -\frac{2}{3}, \][/tex]
[tex]\[ \left( -\frac{2}{3} \right)^2 = \frac{4}{9}. \][/tex]
- Add and subtract [tex]\( \frac{4}{9} \)[/tex] inside the parentheses:
[tex]\[
f(x) = 3 \left( x^2 - \frac{4}{3}x + \frac{4}{9} - \frac{4}{9} \right) + 9,
\][/tex]
[tex]\[
f(x) = 3 \left( \left( x - \frac{2}{3} \right)^2 - \frac{4}{9} \right) + 9.
\][/tex]
4. Simplify the expression:
- Distribute the 3 back into the parentheses:
[tex]\[
f(x) = 3 \left( x - \frac{2}{3} \right)^2 - 3 \left( \frac{4}{9} \right) + 9,
\][/tex]
[tex]\[
f(x) = 3 \left( x - \frac{2}{3} \right)^2 - \frac{12}{9} + 9,
\][/tex]
[tex]\[
f(x) = 3 \left( x - \frac{2}{3} \right)^2 - \frac{4}{3} + 9.
\][/tex]
5. Combine the constants:
- Rewrite [tex]\( 9 \)[/tex] as a fraction with the same denominator to combine with [tex]\( -\frac{4}{3} \)[/tex]:
[tex]\[
9 = \frac{27}{3},
\][/tex]
[tex]\[
f(x) = 3 \left( x - \frac{2}{3} \right)^2 + \left( \frac{27}{3} - \frac{4}{3} \right),
\][/tex]
[tex]\[
f(x) = 3 \left( x - \frac{2}{3} \right)^2 + \frac{23}{3}.
\][/tex]
So, the quadratic function [tex]\( f(x) = 3x^2 - 4x + 9 \)[/tex] in vertex form is:
[tex]\[
f(x) = 3 \left( x - \frac{2}{3} \right)^2 + \frac{23}{3}.
\][/tex]
In this vertex form, [tex]\( a = 3 \)[/tex], [tex]\( h = \frac{2}{3} \)[/tex], and [tex]\( k = \frac{23}{3} \)[/tex].
