Question

Given the table of values, complete the table for the quadratic equation y=−6x^2+3x+8
to find the values of the high or low point of the parabola.

x y
−1
−1
−0.75
2.375
−0.5
5
−0.25

0 8
0.25 8.375
0.5
0.75
1 5
Which of the following is correct?

(1 point)
Responses

(−1,−1)
is the low point on the parabola.
left parenthesis negative 1 comma negative 1 right parenthesis is the low point on the parabola.

(0.5,9)
is the high point on the parabola.
left parenthesis 0.5 comma 9 right parenthesis is the high point on the parabola.

(0.25,8.375)
is the low point on the parabola.
left parenthesis 0.25 comma 8.375 right parenthesis is the low point on the parabola.

(0.25,8.375)
is the high point on the parabola.

Answer 1

To determine whether a point is a high or low point on the parabola represented by the quadratic equation \( y = -6x^2 + 3x + 8 \), we need to find the vertex of the parabola. The vertex gives us the maximum or minimum point, depending on the direction the parabola opens.

1. **Finding the Vertex:**
For a quadratic equation in the form \( y = ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Here, \( a = -6 \) and \( b = 3 \).

Substituting these values into the formula:
\[
x = -\frac{3}{2 \times -6} = -\frac{3}{-12} = \frac{1}{4} = 0.25
\]

2. **Calculating the y-coordinate at \( x = 0.25 \):**
Now we can find the y-coordinate by substituting \( x = 0.25 \) back into the original equation:
\[
y = -6(0.25)^2 + 3(0.25) + 8
\]
\[
= -6(0.0625) + 0.75 + 8
\]
\[
= -0.375 + 0.75 + 8
\]
\[
= 0.375 + 8 = 8.375
\]

3. **Vertex of the Parabola:**
The vertex of the parabola is at \( (0.25, 8.375) \). Since the coefficient of \( x^2 \) (which is \(-6\)) is negative, this means the parabola opens downward; hence, the vertex represents the high point of the parabola.

### Conclusion:
Looking at the options provided:
- \( (0.25, 8.375) \) is the high point of the parabola.

**Correct Response:**
\( (0.25, 8.375) \) is the high point on the parabola.