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

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

(0.25,8.375) is the low point on the parabola.

Answer 1

To determine the high or low point (vertex) of the parabola defined by the quadratic equation \( y = -6x^2 + 3x + 8 \), we can analyze the given equation. Since the quadratic term has a negative coefficient (\(-6\)), we know the parabola opens downward, indicating that the vertex will be the high point.

To find the x-coordinate of the vertex, we can use the formula:

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

where \( a = -6 \) and \( b = 3 \).

Substituting these values into the formula:

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

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

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

Thus, the vertex of the parabola is at \( (0.25, 8.375) \), which is the high point because the parabola opens downward.

From the options given:

1. \( (0.5, 9) \) is the high point on the parabola.
2. \( (-1, -1) \) is the low point on the parabola.
3. \( (0.25, 8.375) \) is the high point on the parabola.
4. \( (0.25, 8.375) \) is the low point on the parabola.

The correct response is:

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