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.

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

1 answer

To analyze the quadratic equation \( y = -6x^2 + 3x + 8 \) and determine the vertex, we first note that this is a downward-facing parabola since the coefficient of \( x^2 \) (which is -6) is negative. The vertex of a parabola in the form \( y = ax^2 + bx + c \) is given by the formula:

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

Here, \( a = -6 \) and \( b = 3 \).

Substituting in:

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

Next, to find the corresponding \( y \)-value of the vertex, we substitute \( x = \frac{1}{4} \) back into the equation:

\[ y = -6\left(\frac{1}{4}\right)^2 + 3\left(\frac{1}{4}\right) + 8 \]

Calculating this step-by-step:

\[ y = -6\left(\frac{1}{16}\right) + \frac{3}{4} + 8 \]

\[ y = -\frac{6}{16} + \frac{3}{4} + 8 \]

\[ y = -\frac{3}{8} + \frac{6}{8} + 8 \]

\[ y = \frac{3}{8} + 8 = \frac{3}{8} + \frac{64}{8} = \frac{67}{8} = 8.375 \]

So, the vertex (the low point) of the parabola is:

\[ \left(\frac{1}{4}, 8.375\right) \]

Now, since this is a downward-facing parabola, it has a maximum point at the vertex, not a minimum. Therefore, we can summarize:

  • The low point on the parabola is \((\frac{1}{4}, 8.375)\).
  • The high point does not exist in this case, as the vertex represents the maximum point.

In conclusion, based on your statements, here are the final values:

  • The vertex is \((\frac{1}{4}, 8.375)\), and this is the high point for the equation \( y = -6x^2 + 3x + 8 \). The values \( (-1, -1) \) and \( (0.5, 9) \) listed in your question seem to be unrelated to this function either mathematically or contextually since they do not match the vertex or corresponding calculations.
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