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 low point on the parabola.
left parenthesis 0.25 comma 8.375 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.
(−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.

Answer 1

To find the vertex of the quadratic equation \( y = -6x^2 + 3x + 8 \), we will first calculate the x-coordinate of the vertex using the formula:

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

In our equation, \( a = -6 \) and \( b = 3 \). Plugging in these values:

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

Next, we substitute \( x = 0.25 \) back into the equation to find the corresponding y-coordinate:

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

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

Since the coefficient of \( x^2 \) in your quadratic equation is negative, the parabola opens downward, meaning the vertex represents the **high point** of the parabola.

Now, let's fill in the missing values in the table by calculating \( y \) for \( x = -0.25 \), \( 0.5 \), and \( 0.75 \):

1. For \( x = -0.25 \):
\[
y = -6(-0.25)^2 + 3(-0.25) + 8
\]
\[
y = -6(0.0625) - 0.75 + 8
\]
\[
y = -0.375 - 0.75 + 8
\]
\[
y = -1.125 + 8 = 6.875
\]

2. For \( x = 0.5 \):
\[
y = -6(0.5)^2 + 3(0.5) + 8
\]
\[
y = -6(0.25) + 1.5 + 8
\]
\[
y = -1.5 + 1.5 + 8
\]
\[
y = 0 + 8 = 8
\]

3. For \( x = 0.75 \):
\[
y = -6(0.75)^2 + 3(0.75) + 8
\]
\[
y = -6(0.5625) + 2.25 + 8
\]
\[
y = -3.375 + 2.25 + 8
\]
\[
y = -1.125 + 8 = 6.875
\]

Now we can summarize the completed values in the table:

| 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 |

Based on this analysis:

- The point \( (0.25, 8.375) \) is the **high point** of the parabola.

The correct response from the options provided is:

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