Question

use the quadratic function ( f(x)=\frac{1}{3}x^2 + 6x + 4 ) to answer the following questions.
a) use the vertex formula to determine the vertex.
the vertex is (square).
(type an ordered pair. simplify your answer.)

Explanation:

Step1: Recall vertex formula for quadratic function

For a quadratic function \( f(x) = ax^2 + bx + c \), the x - coordinate of the vertex is given by \( x = -\frac{b}{2a} \). In the function \( f(x)=\frac{1}{3}x^{2}+6x + 4 \), we have \( a=\frac{1}{3} \) and \( b = 6 \).

Step2: Calculate the x - coordinate of the vertex

Substitute \( a=\frac{1}{3} \) and \( b = 6 \) into the formula \( x=-\frac{b}{2a} \).
\( x=-\frac{6}{2\times\frac{1}{3}}=-\frac{6}{\frac{2}{3}}=-6\times\frac{3}{2}=-9 \)

Step3: Calculate the y - coordinate of the vertex

Substitute \( x = - 9 \) into the function \( f(x)=\frac{1}{3}x^{2}+6x + 4 \).
\( f(-9)=\frac{1}{3}\times(-9)^{2}+6\times(-9)+4=\frac{1}{3}\times81-54 + 4=27-54 + 4=-23 \)

Answer:

\((-9,-23)\)