Certainly! Let’s analyze the function [tex]\( f(x) = -3x^2 - 3x + 6 \)[/tex] step by step to understand its graph and make accurate statements about it.
1. Direction of the Parabola:
- The coefficient of [tex]\( x^2 \)[/tex] is -3, which is negative.
- A quadratic function with a negative [tex]\( x^2 \)[/tex] coefficient opens downwards.
- Statement: The parabola opens downwards.
2. Vertex of the Parabola:
- The vertex form of a quadratic function can be found using the formula for the x-coordinate of the vertex [tex]\( x = -\frac{b}{2a} \)[/tex].
Here, [tex]\( a = -3 \)[/tex] and [tex]\( b = -3 \)[/tex], so
[tex]\[
x = -\frac{-3}{2(-3)} = \frac{3}{-6} = -\frac{1}{2}
\][/tex]
- To find the y-coordinate of the vertex, substitute [tex]\( x = -\frac{1}{2} \)[/tex] back into the function:
[tex]\[
f\left(-\frac{1}{2}\right) = -3\left(-\frac{1}{2}\right)^2 - 3\left(-\frac{1}{2}\right) + 6 = -3\left(\frac{1}{4}\right) + \frac{3}{2} + 6 = -\frac{3}{4} + \frac{3}{2} + 6 = -\frac{3}{4} + \frac{6}{4} + \frac{24}{4} = \frac{27}{4} - \frac{3}{4} = \frac{24}{4} = 6
\][/tex]
- So the vertex of the parabola is [tex]\( \left(-\frac{1}{2}, \frac{27}{4}\right) \)[/tex].
- Statement: The vertex is at [tex]\( \left(-\frac{1}{2}, \frac{27}{4}\right) \)[/tex].
3. Intercepts:
- Y-intercept: The y-intercept is where the function crosses the y-axis (i.e., when [tex]\( x = 0 \)[/tex]):
[tex]\[
f(0) = -3(0)^2 - 3(0) + 6 = 6
\][/tex]
- Statement: The y-intercept is 6.
- X-intercepts: To find the x-intercepts, we solve the quadratic equation:
[tex]\[
-3x^2 - 3x + 6 = 0
\][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = -3 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = 6 \)[/tex]:
[tex]\[
x = \frac{3 \pm \sqrt{(-3)^2 - 4(-3)(6)}}{2(-3)} = \frac{3 \pm \sqrt{9 + 72}}{-6} = \frac{3 \pm \sqrt{81}}{-6} = \frac{3 \pm 9}{-6}
\][/tex]
This gives us two solutions:
[tex]\[
x = \frac{3 + 9}{-6} = \frac{12}{-6} = -2 \quad \text{and} \quad x = \frac{3 - 9}{-6} = \frac{-6}{-6} = 1
\][/tex]
- Statement: The x-intercepts are -2 and 1.
To summarize, the three statements that describe the graph of this function are:
1. The parabola opens downwards.
2. The vertex is at [tex]\( \left(-\frac{1}{2}, \frac{27}{4}\right) \)[/tex].
3. The y-intercept is 6.
Additionally, the x-intercepts are -2 and 1.
