To determine the range of the function [tex]\( f(x) = 3x^2 + 18x \)[/tex], we need to find the minimum or maximum values that the function can take. This involves examining the vertex of the parabola represented by the function. Here's the detailed step-by-step solution:
1. Identify the coefficients:
The function is in the standard form [tex]\( f(x) = ax^2 + bx + c \)[/tex], where [tex]\( a = 3 \)[/tex], [tex]\( b = 18 \)[/tex], and [tex]\( c = 0 \)[/tex].
2. Determine the direction of the parabola:
Since the coefficient of [tex]\( x^2 \)[/tex] (i.e., [tex]\( a \)[/tex]) is positive ([tex]\( a = 3 \)[/tex]), the parabola opens upwards. This implies that the function has a minimum value (vertex) and goes to infinity.
3. Find the x-coordinate of the vertex:
The x-coordinate of the vertex for a parabola in the form [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Substituting [tex]\( a = 3 \)[/tex] and [tex]\( b = 18 \)[/tex],
[tex]\[
x = -\frac{18}{2 \cdot 3} = -\frac{18}{6} = -3
\][/tex]
Therefore, the x-coordinate of the vertex is [tex]\( x = -3 \)[/tex].
4. Calculate the y-coordinate of the vertex:
Substitute [tex]\( x = -3 \)[/tex] into the function to find the y-coordinate,
[tex]\[
f(-3) = 3(-3)^2 + 18(-3)
\][/tex]
[tex]\[
= 3 \cdot 9 - 54
\][/tex]
[tex]\[
= 27 - 54
\][/tex]
[tex]\[
= -27
\][/tex]
So, the y-coordinate of the vertex is [tex]\( -27 \)[/tex].
5. Determine the range:
Since the parabola opens upwards and the minimum value of the function is at [tex]\( y = -27 \)[/tex], the range of the function is all values [tex]\( y \)[/tex] such that [tex]\( y \geq -27 \)[/tex].
Therefore, the range of the function [tex]\( f(x) = 3x^2 + 18x \)[/tex] is [tex]\([-27, \infty)\)[/tex].
Given the choices:
A. [tex]\((- \infty, \infty)\)[/tex]
B. [tex]\([-6, \infty)\)[/tex]
C. [tex]\((- \infty, 18]\)[/tex]
D. [tex]\([-27, \infty)\)[/tex]
The correct answer is:
D. [tex]\([-27, \infty)\)[/tex]
