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.

This table shows the corresponding values of x and y, with some missing values for y.
Steps Solution
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

(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.

(−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 high point on the parabola.

Answer 1

To determine the high or low point of the parabola represented by the equation \(y = -6x^2 + 3x + 8\), we can find the vertex of the parabola. Since the coefficient of \(x^2\) is negative, this parabola opens downwards, meaning it has a maximum (high point) at its vertex.

The vertex of a quadratic equation in standard form \(y = ax^2 + bx + c\) can be found using the formula:

\[
x = -\frac{b}{2a}
\]

In your case, \(a = -6\) and \(b = 3\). Plugging in these values gives:

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

Now, we'll find the corresponding \(y\) value at \(x = 0.25\):

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

Thus, the vertex of the parabola, or the highest point, is \((0.25, 8.375)\).

Now, let's fill in the missing values in the table using the given equation.

1. For \(x = -1\):

\[
y = -6(-1)^2 + 3(-1) + 8 = -6(1) - 3 + 8 = -6 - 3 + 8 = -1
\]

2. For \(x = -0.25\):

\[
y = -6(-0.25)^2 + 3(-0.25) + 8 = -6(0.0625) - 0.75 + 8 = -0.375 - 0.75 + 8 = 6.875
\]

3. For \(x = 0.5\):

\[
y = -6(0.5)^2 + 3(0.5) + 8 = -6(0.25) + 1.5 + 8 = -1.5 + 1.5 + 8 = 8
\]

4. For \(x = 0.75\):

\[
y = -6(0.75)^2 + 3(0.75) + 8 = -6(0.5625) + 2.25 + 8 = -3.375 + 2.25 + 8 = 6.875
\]

Now, the completed table is as follows:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-1 & -1 \\
-0.75 & 2.375 \\
-0.5 & 5 \\
-0.25 & 6.875 \\
0 & 8 \\
0.25 & 8.375 \\
0.5 & 8 \\
0.75 & 6.875 \\
1 & 5 \\
\hline
\end{array}
\]

### Conclusion:
The point \((0.25, 8.375)\) is indeed the highest point (max) of the parabola, making the correct response:

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