Graphs of Quadratic Equations Quick Check

Question

2 of 52 of 5 Items
Question
Given the table of values, complete the table for the quadratic equation y=−6x2+3x+8
y
=
−
6
x
2
+
3
x
+
8
to find the values of the high or low point of the parabola.

x y
−1
−
1
−1
−
1
−0.75
−
0.75
2.375
−0.5
−
0.5
5
−0.25
−
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)
(
−
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.

(0.25,8.375)
(
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.

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

Answer 1

To find the high or low point of the parabola represented by the quadratic equation \( y = -6x^2 + 3x + 8 \), we need to determine whether the vertex of the parabola is a maximum (high point) or minimum (low point).

The vertex \( x \)-coordinate for a quadratic equation in the form \( y = ax^2 + bx + c \) is given by the formula:

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

In this equation:

• \( a = -6 \)
• \( b = 3 \)

Calculating the \( x \)-coordinate of the vertex:

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

Next, we substitute \( x = 0.25 \) back into the equation to find the corresponding \( y \)-value:

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

Calculating each term:

1. \( (0.25)^2 = 0.0625 \)
2. \( -6 \cdot 0.0625 = -0.375 \)
3. \( 3 \cdot 0.25 = 0.75 \)

Putting it all together:

\[ y = -0.375 + 0.75 + 8 = 0.375 + 8 = 8.375 \]

Thus, the vertex is \( (0.25, 8.375) \), and since \( a < 0 \) (the parabola opens downward), this point is the high point of the parabola.

Based on the available choices, the correct response is:

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