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Which of the following is a point you can generate to create the parabola for the quadratic equation y=−6x^2+3x+8 ?(1 point) Responses (−6,206) left parenthesis negative 6 comma 206 right parentheis (−3,−55) left parenthesis negative 3 comma negative 55 right parentheis (−3,323) left parenthesis negative 3 comma 323 right parentheis (−6,8)

3 answers

To determine which of the given points lies on the parabola defined by the quadratic equation \(y = -6x^2 + 3x + 8\), we can substitute each of the provided \(x\) values into the equation and check if the resulting \(y\) value matches the \(y\) value in the point.

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

  2. For the point \((-3, -55)\): \[ y = -6(-3)^2 + 3(-3) + 8 = -6(9) - 9 + 8 = -54 - 9 + 8 = -55 \quad \text{(a match)} \]

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

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

The only point that lies on the parabola is \((-3, -55)\).

Given the table of values, complete the table for the quadratic equation y=−6x^2+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? (1 point) Responses (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) 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. (−1,−1) is the low point on the parabola.

To find the coordinates of the high or low point (vertex) of the parabola described by the quadratic equation \(y = -6x^2 + 3x + 8\), we can analyze the provided points.

The vertex form of a quadratic equation \(y = ax^2 + bx + c\) has its vertex (high/low point) at the \(x\) value given by the formula \(x = -\frac{b}{2a}\).

For the equation \(y = -6x^2 + 3x + 8\):

  • \(a = -6\)
  • \(b = 3\)

Calculating the \(x\) value of the vertex: \[ x = -\frac{b}{2a} = -\frac{3}{2(-6)} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]

Next, we substitute \(x = 0.25\) back into the equation to find the corresponding \(y\): \[ y = -6(0.25)^2 + 3(0.25) + 8 = -6(0.0625) + 0.75 + 8 = -0.375 + 0.75 + 8 = 8.375 \]

Thus, the vertex of the parabola is at \((0.25, 8.375)\), which is indeed a high point because the parabola opens downwards (as indicated by the negative coefficient of \(x^2\)).

Now, we can evaluate the options:

  • \((0.25, 8.375)\) is the low point on the parabola. (Incorrect)
  • \((0.5, 9)\) is the high point on the parabola. (Incorrect)
  • \((0.25, 8.375)\) is the high point on the parabola. (Correct)
  • \((-1, -1)\) is the low point on the parabola. (Incorrect)

Thus, the correct answer is: \[ (0.25, 8.375) \text{ is the high point on the parabola.} \]