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.

Which of the following is correct?

A: ( 0.5, 9) is the high point on the parabola.

B: ( 0.25, 8.375) is the low point on the parabola

C: ( -1, -1) is the low point on the parabola

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

Answer 1

To determine the high or low point (the vertex) of the parabola given by the quadratic equation \( y = -6x^2 + 3x + 8 \), we can use the vertex formula. The x-coordinate of the vertex for a quadratic equation of the form \( y = ax^2 + bx + c \) is found using:

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

In our equation, \( a = -6 \), \( b = 3 \), and \( c = 8 \).

Let's calculate the x-coordinate of the vertex:

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

Next, we need to calculate the y-coordinate of the vertex by substituting \( x = 0.25 \) back into the original equation:

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

Calculating \( (0.25)^2 \):

\[
(0.25)^2 = 0.0625
\]

Substituting that into the equation:

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

So the vertex, which is the high point of the parabola (because the parabola opens downwards, indicated by a negative \( a \)), is at:

\[
(0.25, 8.375)
\]

Therefore, let's evaluate the options given:

A: (0.5, 9) is the high point on the parabola. - **Incorrect**.

B: (0.25, 8.375) is the low point on the parabola. - **Incorrect**, it's a high point.

C: (-1, -1) is the low point on the parabola. - **Incorrect**.

D: (0.25, 8.375) is the high point on the parabola. - **Correct**.

The correct option is **D: (0.25, 8.375) is the high point on the parabola.**