Question

Given the table of values, complete the table for the quadratic equation y=−6x2+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.

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

Answer 1

To find the high or low point (vertex) of the quadratic equation \( y = -6x^2 + 3x + 8 \), we first need to calculate all the given values and then identify the vertex.

### Calculation of Missing Values

1. **For \( x = -0.25 \):**
\[
y = -6(-0.25)^2 + 3(-0.25) + 8 = -6(0.0625) - 0.75 + 8 = -0.375 - 0.75 + 8 = 6.875
\]

2. **For \( x = 0.5 \):**
\[
y = -6(0.5)^2 + 3(0.5) + 8 = -6(0.25) + 1.5 + 8 = -1.5 + 1.5 + 8 = 8
\]

3. **For \( x = 0.75 \):**
\[
y = -6(0.75)^2 + 3(0.75) + 8 = -6(0.5625) + 2.25 + 8 = -3.375 + 2.25 + 8 = 6.875
\]

The completed table is as follows:

```
x | y
---------------
−1 | −1
−0.75 | 2.375
−0.5 | 5
−0.25 | 6.875
0 | 8
0.25 | 8.375
0.5 | 8
0.75 | 6.875
1 | 5
```

### Determining the Vertex

For a quadratic function of the form \( ax^2 + bx + c \):
- The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \).
- Here, \( a = -6 \) and \( b = 3 \).

\[
x = -\frac{3}{2(-6)} = \frac{3}{12} = 0.25
\]

### Finding the y-coordinate of the vertex

Substituting \( x = 0.25 \) into the original equation to find \( y \):

\[
y = -6(0.25)^2 + 3(0.25) + 8 = -6(0.0625) + 0.75 + 8 = -0.375 + 0.75 + 8 = 8.375
\]

### Conclusion

The vertex, \( (0.25, 8.375) \), is the **high point** of the parabola because the coefficient of \( x^2 \) is negative (indicating a downward-opening parabola).

From the options provided:

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

This response is correct:

\[
\text{(0.25,8.375) is the high point on the parabola.}
\]