Question

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

(-3, -55)
(-3,323)
(-6,8)
(-6,206)

Answer 1

To analyze the parabola given by the equation \( y = -6x^2 + 3x + 8 \), it's helpful to understand its properties, such as its vertex, direction, and y-intercept.

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

### 2. **Y-Intercept**:
The y-intercept can be found by setting \( x = 0 \):

\[
y = -6(0)^2 + 3(0) + 8 = 8
\]

So, the y-intercept is at the point \( (0, 8) \).

### 3. **Vertex**:
The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \):

\[
a = -6, \quad b = 3 \quad \Rightarrow \quad x = -\frac{3}{2(-6)} = \frac{3}{12} = \frac{1}{4}
\]

Now, substitute \( x = \frac{1}{4} \) back into the equation to find the \( y \)-coordinate of the vertex:

\[
y = -6\left(\frac{1}{4}\right)^2 + 3\left(\frac{1}{4}\right) + 8
\]
\[
= -6\left(\frac{1}{16}\right) + \frac{3}{4} + 8
\]
\[
= -\frac{6}{16} + \frac{3}{4} + 8
\]
\[
= -\frac{3}{8} + \frac{6}{8} + 8
\]
\[
= -\frac{3}{8} + \frac{6}{8} + \frac{64}{8}
\]
\[
= \frac{-3 + 6 + 64}{8} = \frac{67}{8}
\]

So, the vertex is at \( \left( \frac{1}{4}, \frac{67}{8} \right) \).

### 4. **Finding Specific Points**:
Now we can check the points given in the options to see if any of them lie on the parabola.

1. For \( (-3, -55) \):
\[
y = -6(-3)^2 + 3(-3) + 8 = -6(9) - 9 + 8 = -54 - 9 + 8 = -55 \quad \text{(This point is on the parabola)}
\]

2. For \( (-3, 323) \):
This value for y cannot be viable.

3. For \( (-6, 8) \):
\[
y = -6(-6)^2 + 3(-6) + 8 = -6(36) - 18 + 8 = -216 - 18 + 8 = -226 \quad \text{(not on the parabola)}
\]

4. For \( (-6, 206) \):
This value for y cannot be viable.

### Conclusion:
The only point that lies on the parabola defined by the equation \( y = -6x^2 + 3x + 8 \) is the point \((-3, -55)\).