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

(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.

(−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 low point on the parabola.

Answer 1

To find the high or low point (vertex) of the parabola defined by the quadratic equation \(y = -6x^2 + 3x + 8\), we can use the formula for the x-coordinate of the vertex of a parabola given by \(x = -\frac{b}{2a}\), where \(a\) and \(b\) are the coefficients from the equation \(y = ax^2 + bx + c\).

In this case:
- \(a = -6\)
- \(b = 3\)
- \(c = 8\)

Calculating the x-coordinate of the vertex:
\[
x = -\frac{3}{2 \cdot -6} = -\frac{3}{-12} = \frac{1}{4} = 0.25
\]

Next, we substitute \(x = 0.25\) back into the equation to find the y-coordinate of the vertex:
\[
y = -6(0.25)^2 + 3(0.25) + 8
\]
Calculating:
\[
= -6(0.0625) + 0.75 + 8
= -0.375 + 0.75 + 8
= 0.375 + 8
= 8.375
\]

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

Since the leading coefficient \(a\) is negative (-6), the parabola opens downward, indicating that the vertex represents a high point.

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