Question

Which of the following is a point you can generate to create the parabola for the quadratic equation y=−6x2+3x+8

Responses

(−6,206)

(−6,8)

(−3,−55)

(−3,323)

Answer 1

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.

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?


Responses

(0.5,9)


(0.25,8.375)


(−1,−1)


(0.25,8.375)

Answer 2

To determine if any of the given points lie on the parabola described by the equation \(y = -6x^2 + 3x + 8\), we can substitute the x-values from the responses into the equation and check if the corresponding y-values are correct.

1. **For the point (−6, 206)**:
\[
y = -6(-6)^2 + 3(-6) + 8 = -6(36) - 18 + 8 = -216 - 18 + 8 = -226 \quad (\text{not } 206)
\]

2. **For the point (−6, 8)**:
\[
y = -6(-6)^2 + 3(-6) + 8 = -6(36) - 18 + 8 = -216 - 18 + 8 = -226 \quad (\text{not } 8)
\]

3. **For the point (−3, −55)**:
\[
y = -6(-3)^2 + 3(-3) + 8 = -6(9) - 9 + 8 = -54 - 9 + 8 = -55 \quad (\text{this one works!})
\]

4. **For the point (−3, 323)**:
\[
y = -6(-3)^2 + 3(-3) + 8 = -6(9) - 9 + 8 = -54 - 9 + 8 = -55 \quad (\text{not } 323)
\]

From these calculations, the point that lies on the parabola defined by the quadratic equation \(y = -6x^2 + 3x + 8\) is:

**(−3, −55)**.

Answer 3

To complete the table for the quadratic equation \(y = -6x^2 + 3x + 8\), we will evaluate the function \(y\) for the missing \(x\) values.

### Given data:
1. \(x = -1\):
\[
y = -6(-1)^2 + 3(-1) + 8 = -6(1) - 3 + 8 = -6 - 3 + 8 = -1
\]
So, \((-1, -1)\) completes the table.

2. \(x = -0.75\):
\[
y = -6(-0.75)^2 + 3(-0.75) + 8 = -6(0.5625) - 2.25 + 8 = -3.375 - 2.25 + 8 = 2.375
\]
So, \((-0.75, 2.375)\) completes the table.

3. \(x = -0.5\):
\[
y = -6(-0.5)^2 + 3(-0.5) + 8 = -6(0.25) - 1.5 + 8 = -1.5 - 1.5 + 8 = 5
\]
So, \((-0.5, 5)\) completes the table.

4. \(x = -0.25\):
\[
y = -6(-0.25)^2 + 3(-0.25) + 8 = -6(0.0625) - 0.75 + 8 = -0.375 - 0.75 + 8 = 6.875
\]
So, \((-0.25, 6.875)\) completes the table.

5. \(x = 0.5\):
\[
y = -6(0.5)^2 + 3(0.5) + 8 = -6(0.25) + 1.5 + 8 = -1.5 + 1.5 + 8 = 8
\]
So, \((0.5, 8)\) completes the table.

6. \(x = 0.75\):
\[
y = -6(0.75)^2 + 3(0.75) + 8 = -6(0.5625) + 2.25 + 8 = -3.375 + 2.25 + 8 = 6.875
\]
So, \((0.75, 6.875)\) completes the table.

7. \(x = 1\):
\[
y = -6(1)^2 + 3(1) + 8 = -6 + 3 + 8 = 5
\]
So, \((1, 5)\) is already provided.

### Completed Table
- For \(x = -1\), \(y = -1\).
- For \(x = -0.75\), \(y = 2.375\).
- For \(x = -0.5\), \(y = 5\).
- For \(x = -0.25\), \(y = 6.875\).
- For \(x = 0\), \(y = 8\).
- For \(x = 0.25\), \(y = 8.375\).
- For \(x = 0.5\), \(y = 8\).
- For \(x = 0.75\), \(y = 6.875\).
- For \(x = 1\), \(y = 5\).

### Finding the High or Low Point
Since the quadratic opens downwards (the coefficient of \(x^2\) is negative), the vertex will be the maximum point (high point) of the parabola.

To find the vertex \(x\)-coordinate:
\[
x = -\frac{b}{2a} = -\frac{3}{2(-6)} = \frac{3}{12} = 0.25
\]
Now plug \(x = 0.25\) back into the equation to find \(y\):
\[
y = -6(0.25)^2 + 3(0.25) + 8 = -6(0.0625) + 0.75 + 8 = -0.375 + 0.75 + 8 = 8.375
\]
So, the vertex (high point) is \((0.25, 8.375)\).

### Response Conclusion
Among the responses given:
- **(0.25, 8.375)** is the correct one.