Question

What is the parabola for y = -6x^2 + 3x + 8?

Answer 1

To analyze the parabola given by the equation \(y = -6x^2 + 3x + 8\), we can identify important characteristics such as the vertex, direction, and intercepts.

1. **Direction**: Since the coefficient of \(x^2\) is negative (-6), the parabola opens downward.

2. **Vertex**: The vertex of a parabola in the form \(y = ax^2 + bx + c\) can be found using the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
In this case, \(a = -6\) and \(b = 3\):
\[
x = -\frac{3}{2(-6)} = \frac{3}{12} = \frac{1}{4}
\]

Now, substitute \(x = \frac{1}{4}\) back into the equation to find the y-coordinate of the vertex:
\[
y = -6\left(\frac{1}{4}\right)^2 + 3\left(\frac{1}{4}\right) + 8
\]
\[
y = -6\left(\frac{1}{16}\right) + \frac{3}{4} + 8
\]
\[
y = -\frac{6}{16} + \frac{3}{4} + 8 = -\frac{3}{8} + \frac{6}{8} + 8 = \frac{3}{8} + 8
\]
Converting 8 to eighths gives:
\[
8 = \frac{64}{8}
\]
Hence,
\[
y = \frac{3}{8} + \frac{64}{8} = \frac{67}{8}
\]
The vertex of the parabola is \(\left(\frac{1}{4}, \frac{67}{8}\right)\).

3. **Y-intercept**: To find the y-intercept, set \(x = 0\):
\[
y = -6(0)^2 + 3(0) + 8 = 8
\]
So the y-intercept is \( (0, 8) \).

4. **X-intercepts**: To find the x-intercepts, set \(y = 0\):
\[
0 = -6x^2 + 3x + 8
\]
Rearranging gives:
\[
6x^2 - 3x - 8 = 0
\]
We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 6\), \(b = -3\), and \(c = -8\):
\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(6)(-8)}}{2(6)}
\]
\[
x = \frac{3 \pm \sqrt{9 + 192}}{12}
\]
\[
x = \frac{3 \pm \sqrt{201}}{12}
\]

Thus, the x-intercepts are:
\[
x = \frac{3 + \sqrt{201}}{12}, \quad x = \frac{3 - \sqrt{201}}{12}
\]

### Summary:
- The parabola opens downward.
- Vertex: \(\left(\frac{1}{4}, \frac{67}{8}\right)\)
- Y-intercept: \((0, 8)\)
- X-intercepts: \(x = \frac{3 \pm \sqrt{201}}{12}\)

This analysis gives you a comprehensive view of the parabola represented by the equation \(y = -6x^2 + 3x + 8\).