To determine the quadrant of the vertex of the given quadratic function [tex]\( f(x) = 3x^2 + 30x + 60 \)[/tex], we need to follow these steps:
1. Identify the coefficients of the quadratic function:
In the function [tex]\( f(x) = 3x^2 + 30x + 60 \)[/tex], the coefficients are:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 30 \)[/tex]
- [tex]\( c = 60 \)[/tex]
2. Determine the x-coordinate of the vertex:
The x-coordinate of the vertex for a quadratic equation [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Plugging in the given coefficients:
[tex]\[
x = -\frac{30}{2 \times 3} = -\frac{30}{6} = -5
\][/tex]
3. Determine the y-coordinate of the vertex:
We find the y-coordinate by substituting [tex]\( x = -5 \)[/tex] back into the original function [tex]\( f(x) \)[/tex]:
[tex]\[
y = 3(-5)^2 + 30(-5) + 60
\][/tex]
[tex]\[
y = 3(25) + 30(-5) + 60
\][/tex]
[tex]\[
y = 75 - 150 + 60
\][/tex]
[tex]\[
y = 75 - 150 + 60 = -15
\][/tex]
4. Identify the quadrant:
The vertex of the function is at the point [tex]\((-5, -15)\)[/tex]. To determine in which quadrant this point lies:
- The x-coordinate [tex]\(-5\)[/tex] is negative.
- The y-coordinate [tex]\(-15\)[/tex] is also negative.
A point with negative x and negative y coordinates is located in Quadrant III of the Cartesian plane.
Therefore, the answer is:
(C) Quadrant III
