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.

Question

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

(−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.

(0.5,9) is the high point on the parabola.
left parenthesis 0.5 comma 9 right parenthesis is the high point on the parabola.

(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.

(0.25,8.375) is the high point on the parabola.
left parenthesis 0.25 comma 8.375 right parenthesis is the high point on the parabola.
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Answer 1

To determine the high or low point of the parabola described by the quadratic equation \( y = -6x^2 + 3x + 8 \), we need to analyze the equation.

1. Identifying the vertex: In a quadratic equation of the form \( y = ax^2 + bx + c \), the x-coordinate of the vertex (which gives the location of the high or low point depending on the direction the parabola opens) can be found using the formula: \[ x = -\frac{b}{2a} \] Here, \( a = -6 \) and \( b = 3 \). \[ x = -\frac{3}{2 \times -6} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]

2. Calculating the y-coordinate at \( x = 0.25 \): Substitute \( x = 0.25 \) into the equation to find the corresponding y-coordinate: \[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ = -6(0.0625) + 0.75 + 8 \] \[ = -0.375 + 0.75 + 8 \] \[ = 8.375 \]

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

3. Analysis of the vertex: Since the coefficient \( a \) is negative (\( -6 \)), the parabola opens downwards, which means that the vertex is the high point of the parabola.

Given the analysis above, the correct response regarding the various points is:

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