To find the high or low point (the vertex) of the quadratic equation \(y = -6x^2 + 3x + 8\), we can use the formula for the x-coordinate of the vertex, which is given by:
\[ x = -\frac{b}{2a} \]
In this case, \(a = -6\) and \(b = 3\).
Calculating the x-coordinate of the vertex:
\[ x = -\frac{3}{2(-6)} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]
Now we substitute \(x = 0.25\) back into the equation to find the corresponding y-coordinate:
\[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ y = -6(0.0625) + 0.75 + 8 \] \[ y = -0.375 + 0.75 + 8 \] \[ y = 0.375 + 8 = 8.375 \]
The vertex of the parabola is \((0.25, 8.375)\). Because the quadratic opens downwards (since \(a < 0\)), this point is a high point (maximum) of the parabola.
Now, looking at the options given:
- (0.25, 8.375) is the low point on the parabola. (Incorrect)
- (0.5, 9) is the high point on the parabola. (Incorrect)
- (0.25, 8.375) is the high point on the parabola. (Correct)
- (−1, −1) is the low point on the parabola. (Incorrect)
The correct response is:
(0.25, 8.375) is the high point on the parabola.