Question

use the quadratic function ( f(x)=\frac{1}{2}x^2 + 6x + 6 ) 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

For \( f(x)=ax^2 + bx + c \), vertex x - coordinate is \( x = -\frac{b}{2a} \). Here, \( a=\frac{1}{2} \), \( b = 6 \), \( c = 6 \).
\( x=-\frac{6}{2\times\frac{1}{2}}=-\frac{6}{1}=-6 \)

Step2: Find y - coordinate

Substitute \( x=-6 \) into \( f(x)=\frac{1}{2}x^2 + 6x + 6 \).
\( f(-6)=\frac{1}{2}(-6)^2+6(-6)+6=\frac{1}{2}(36)-36 + 6=18-36 + 6=-12 \)

Answer:

\((-6, -12)\)