To describe the graph of the function [tex]\( f(x) = -3x^2 - 3x + 6 \)[/tex], we need to determine a few key characteristics of the graph. Let's start with finding the vertex of the parabola.
1. Determine the vertex:
The general form of a quadratic function is [tex]\( ax^2 + bx + c \)[/tex]. In this case, [tex]\( a = -3 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = 6 \)[/tex].
To find the x-coordinate of the vertex of a parabola given by [tex]\( ax^2 + bx + c \)[/tex], we use the formula:
[tex]\[
x_{\text{vertex}} = -\frac{b}{2a}
\][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
x_{\text{vertex}} = -\frac{-3}{2 \times -3} = -\frac{-3}{-6} = 0.5
\][/tex]
To find the y-coordinate of the vertex, we substitute [tex]\( x = -0.5 \)[/tex] back into the function:
[tex]\[
f(-0.5) = -3(-0.5)^2 - 3(-0.5) + 6
\][/tex]
[tex]\[
f(-0.5) = -3(0.25) + 1.5 + 6
\][/tex]
[tex]\[
f(-0.5) = -0.75 + 1.5 + 6
\][/tex]
[tex]\[
f(-0.5) = 6.75
\][/tex]
Thus, the vertex of the parabola is [tex]\( (-0.5, 6.75) \)[/tex].
2. Determine the nature of the vertex:
Since the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a = -3 \)[/tex]) is negative, the parabola opens downward. This means the vertex represents the maximum value of the function.
Based on the vertex information and the nature of the parabola, we can describe the graph with the following statements:
1. The vertex is [tex]\( (-0.5, 6.75) \)[/tex].
2. The parabola opens downward because the coefficient of [tex]\( x^2 \)[/tex] is negative.
3. The vertex represents the maximum value of the function [tex]\( f(x) \)[/tex].
So, the correct statements that describe the graph are:
1. The vertex is [tex]\((-0.5, 6.75)\)[/tex].
2. The graph has a maximum value at the vertex since the parabola opens downward.
3. The vertex is the maximum value.
These statements accurately describe the graph of the function [tex]\( f(x) = -3x^2 - 3x + 6 \)[/tex].
