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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)

3 answers

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).

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)

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.