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 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 find the vertex of the parabola represented by the quadratic equation $y=-6x^2+3x+8$, we can use the formula for the x-coordinate of the vertex, which is given by:

$$x=-\frac{b}{2a}$$

where $a=-6$ and $b=3$.

Calculating the x-coordinate:

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

Now, we can find the corresponding y-coordinate by substituting $x=0.25$ back into the equation:

$$y=-6(0.25{)}^2+3(0.25)+8$$ $$=-6(0.0625)+0.75+8$$ $$=-0.375+0.75+8$$ $$=0.375+8=8.375$$

Thus, the vertex of the parabola is at the point $(0.25,8.375)$. Since the coefficient of $x^2$ is negative ($-6$), this point is a maximum (high point) of the parabola.

Now to complete the table:

• At $x=-0.25$: $$y=-6(-0.25{)}^2+3(-0.25)+8$$ $$=-6(0.0625)-0.75+8$$ $$=-0.375-0.75+8$$ $$=-1.125+8=6.875$$

• At $x=0.5$: $$y=-6(0.5{)}^2+3(0.5)+8$$ $$=-6(0.25)+1.5+8$$ $$=-1.5+1.5+8=8$$

• At $x=0.75$: $$y=-6(0.75{)}^2+3(0.75)+8$$ $$=-6(0.5625)+2.25+8$$ $$=-3.375+2.25+8=6.875$$

Now the completed table looks like this:

$$\text{Misplaced hline}$$

Summary of Response Options

Given the calculated points, the correct statement from the options provided is:

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

None of the other statements provided are correct based on the calculations. Thus, the relevant option is the one regarding $(0.25,8.375)$ as the high point on the parabola.