Use the quadratic function to determine if the function has a maximum or minimum value and then find this maximum or minimum value.
-$$f ( x ) = x ^ { 2 } + 6 x + 3$$
A) minimum at $( - 3 , - 6 )$
B) minimum at $( 0 , - 3 )$
C) maximum at $( - 6 , - 3 )$
D) maximum at $( - 6,0 )$

Question

Quizplus · Accepted Answer

The quadratic function $f(x) = x^2 + 6x + 3$ is in the form $ax^2 + bx + c$, where $a = 1$, $b = 6$, and $c = 3$. Since $a > 0$, the parabola opens upwards, indicating a minimum value. The vertex of the parabola, which gives the minimum value, can be found using the formula $-\frac{b}{2a}$ for the x-coordinate. Substituting $a = 1$ and $b = 6$ gives $-\frac{6}{2(1)} = -3$. To find the y-coordinate of the vertex, substitute $x = -3$ into the original equation: $f(-3) = (-3)^2 + 6(-3) + 3 = 9 - 18 + 3 = -6$. Therefore, the function has a minimum value at $(-3, -6)$.

Use the Quadratic Function to Determine If the Function Has f(x)=x2+6x+3f ( x ) = x ^ { 2 } + 6 x + 3f(x)=x2+6x+3

Use the Quadratic Function to Determine If the Function Has $f ( x ) = x ^ { 2 } + 6 x + 3$