Question

Given the table of values, complete the table for the quadratic equation y=−6x2+3x+8
𝑦
=
−
6
𝑥
2
+
3
𝑥
+
8
to find the values of the high or low point of the parabola.

x y
−1
−
1
−1
−
1
−0.75
−
0.75
2.375
−0.5
−
0.5
5
−0.25
−
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)
(
−
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)
(
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)
(
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)
(
0.25
,
8.375
)
is the low point on the parabola.

Answer 1

To determine whether the points listed in the provided options are high or low points of the quadratic equation \( y = -6x^2 + 3x + 8 \), we first need to calculate the vertex of the parabola.

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

\[
x = -\frac{3}{2(-6)} = -\frac{3}{-12} = \frac{1}{4} = 0.25
\]

Next, we will substitute \( x = 0.25 \) back into the quadratic equation to find the corresponding \( y \) value:

\[
y = -6(0.25)^2 + 3(0.25) + 8
\]
Calculating \( (0.25)^2 = 0.0625 \):

\[
y = -6(0.0625) + 0.75 + 8
\]

\[
y = -0.375 + 0.75 + 8
\]

\[
y = 8.375
\]

Thus, the vertex point is \( (0.25, 8.375) \). Since the parabola opens downward (as indicated by the negative \( a \) value), this vertex represents the highest point (the maximum) of the parabola.

Now, evaluating the options:

1. **(−1,−1)** is the low point on the parabola. - Incorrect; this point is not a vertex.
2. **(0.25,8.375)** is the high point on the parabola. - Correct; we have found that this is the maximum of the parabola.
3. **(0.5,9)** is the high point on the parabola. - Incorrect; this point does not even correspond to the vertex we found.
4. **(0.25,8.375)** is the low point on the parabola. - Incorrect; we established it as the high point.

Thus, the correct response is:
**(0.25,8.375) is the high point on the parabola.**