Question

determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value.

f(x)=3x^{2}+18x - 9

does the quadratic function f have a minimum value or a maximum value?

the function f has a maximum value.

the function f has a minimum value.

Explanation:

Step1: Identify the coefficient of $x^{2}$

For the quadratic function $f(x)=3x^{2}+18x - 9$, the coefficient $a = 3$.

Step2: Determine max - min

Since $a=3>0$, the parabola opens upward. So the function has a minimum value.

Step3: Find the x - coordinate of the vertex

The x - coordinate of the vertex of a quadratic function $y = ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. Here, $a = 3$ and $b = 18$. So $x=-\frac{18}{2\times3}=- 3$.

Step4: Find the minimum value

Substitute $x=-3$ into the function $f(x)=3x^{2}+18x - 9$. Then $f(-3)=3\times(-3)^{2}+18\times(-3)-9=3\times9-54 - 9=27-54 - 9=-36$.

Answer:

The function f has a minimum value. The minimum value is - 36.