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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

?(1 point)
Responses

(−6,8)
left parenthesis negative 6 comma 8 right parentheis

(−6,206)
left parenthesis negative 6 comma 206 right parentheis

(−3,323)
left parenthesis negative 3 comma 323 right parentheis

(−3,−55)

9 answers

To find a point that lies on the parabola defined by the quadratic equation \( y = -6x^2 + 3x + 8 \), we substitute the x-values from the provided options into the equation and see which one produces a correct corresponding y-value.

Let's test each of the given points:

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

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

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

  4. For \((-3, -55)\): \[ y = -6(-3)^2 + 3(-3) + 8 = -6(9) - 9 + 8 = -54 - 9 + 8 = -55 \quad (\text{this is correct}) \]

The point \((-3, -55)\) is indeed a point on the parabola described by the equation \( y = -6x^2 + 3x + 8 \).

So the correct response 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?

(1 point)
Responses

(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.
left parenthesis 0.25 comma 8.375 right parenthesis is the low point on the parabola.

(−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.
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To find the high or low point of the parabola representing the quadratic equation \( y = -6x^2 + 3x + 8 \), we can first determine that since the leading coefficient (\(-6\)) is negative, the parabola opens downwards. Therefore, the vertex will be the high point of the parabola.

The vertex \( x \)-coordinate for a quadratic in the form \( y = ax^2 + bx + c \) can be calculated using the formula \( x = -\frac{b}{2a} \).

In this case:

  • \( a = -6 \)
  • \( b = 3 \)

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

Now we can find the \( y \)-coordinate at \( x = 0.25 \) by substituting it back into the equation: \[ y = -6(0.25)^2 + 3(0.25) + 8 = -6(0.0625) + 0.75 + 8 = -0.375 + 0.75 + 8 = 8.375 \]

Thus, the vertex (high point) of the parabola is at the point \( (0.25, 8.375) \).

Now, looking at the options provided:

  1. \((0.25, 8.375)\) is the high point on the parabola.
  2. \((0.25, 8.375)\) is the low point on the parabola.
  3. \((-1, -1)\) is the low point on the parabola.
  4. \((0.5, 9)\) is the high point on the parabola.

The correct response is: \((0.25, 8.375)\) is the high point on the parabola.

Use the image to answer the question.

A parabola is graphed on a coordinate plane. The x-axis ranges from negative 4 to 6 in increments of 1. The y-ais ranges from negative 8 to 4 in increments of 1.

Identify one of the zeros (or x-intercepts) on the graph.

(1 point)
Responses

(0,−4)
left parenthesis 0 comma negative 4 right parenthesis

(0,2)
left parenthesis 0 comma 2 right parenthesis

(4,0)
left parenthesis 4 comma 0 right parenthesis

x=1.5

To identify one of the zeros (or x-intercepts) on the graph of a parabola, we need to find a point where the parabola crosses the x-axis. The x-value of this point will indicate the x-intercept, which corresponds to a point with \( y = 0 \).

The options given are:

  1. \( (0, -4) \) - This point is below the x-axis (not a zero).
  2. \( (0, 2) \) - This point is above the x-axis (not a zero).
  3. \( (4, 0) \) - This point is on the x-axis (this is a zero).
  4. \( x = 1.5 \) - This is a vertical line but does not specify a point.

Among the provided options, the point \( (4, 0) \) is on the x-axis, indicating that it is indeed one of the zeros of the parabola.

So, the correct response is: \( (4, 0) \).

Use the image to answer the question.

A parabola is graphed on a coordinate plane. The x-axis ranges from negative 14 to 1 in increments of 1. The y-axis ranges from negative 2 to 6 in increments of 1.

Identify the vertex on the graph.

(1 point)
Responses

x=−11
x equals negative 11

(−12,0)
left parenthesis negative 12 comma 0 right parenthesis

(−10,0)
left parenthesis negative 10 comma 0 right parenthesis

(−11,−1)

To identify the vertex of a parabola from a graph, we need to find the point that represents the highest (for parabolas opening downwards) or lowest (for parabolas opening upwards) point of the curve.

Given the choices:

  1. \( x = -11 \) - This is a vertical line but does not provide the coordinates of a point.
  2. \( (-12, 0) \) - This point is a possible coordinate but does not give enough context about the vertex's position relative to the parabola.
  3. \( (-10, 0) \) - This point is also a possible coordinate but has the same issue as above.
  4. \( (-11, -1) \) - Given that it includes a negative y-value, this could indicate a potential vertex.

Without the visual of the graph, the best approach is to look for logical reasoning.

If the graph shows a downward-opening parabola, the vertex coordinates would likely align with the minimum or maximum values identified in the context, often found near the center of the x-values at their highest or lowest points.

Assuming standard behavior of parabolas and regarding the choices, if the vertex appears to be at a minimum point within the range, then \( (-11, -1) \) seems to aptly suggest that it’s at the bottom of the curve, potentially making it the vertex.

Therefore, the correct response for identifying the vertex is: \( (-11, -1) \).

Use the image to answer the question.

A parabola is graphed on a coordinate plane. The x-axis ranges from negative 5 to 5 in increments of 1. The y-axis ranges from negative 1 to 16 in increments of 1.

Determine which of the following statements is true about the parabola.

(1 point)
Responses

There is only one x-intercept.
There is only one x -intercept.

The y-intercept and the x-intercepts share the same point.
The y -intercept and the x -intercepts share the same point.

The vertex and the x-intercepts share the same point.
The vertex and the x -intercepts share the same point.

The y-intercept and the vertex share the same point.

To determine which statement is true about the parabola based on the description provided, let's evaluate each statement:

  1. There is only one x-intercept.

    • A parabola can have zero, one, or two x-intercepts depending on its orientation and position. If the parabola touches the x-axis at one point (the vertex), it has one x-intercept; if it opens upwards and the vertex is below the x-axis, it could have two x-intercepts.
    • Since we do not have the graph to confirm the number of x-intercepts, this statement cannot be conclusively verified.

  2. The y-intercept and the x-intercepts share the same point.

    • The y-intercept occurs where \( x = 0 \) and the x-intercepts occur where \( y = 0 \). For these two to be the same, the point where the parabola crosses the y-axis must also be where it crosses the x-axis, which is generally not the case for standard parabolas. Thus, this statement is unlikely to be true.

  3. The vertex and the x-intercepts share the same point.

    • The vertex of a parabola can be an x-intercept if the parabola is tangent to the x-axis at that vertex, meaning it only touches the x-axis at that point. However, not all parabolas will have their vertex at an x-intercept point. Thus, this statement is also unlikely to be true in general.

  4. The y-intercept and the vertex share the same point.

    • This is plausible. The vertex and y-intercept can coincide, particularly in parabolas that open upward with the vertex being at the lowest point (for instance, if the vertex is at \( (0, y) \) and the parabola opens upwards).

Based on the evaluation of the statements and considering that both the y-intercept and vertex can indeed occupy the same point, along with regular behavior observed in parabolas, the most plausible statement to be true would be:

The y-intercept and the vertex share the same point.

This seems the most reasonable of the provided options without the actual graph visible.