Question

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 (−3,−55) left parenthesis negative 3 comma negative 55 right parentheis (−6,206) left parenthesis negative 6 comma 206 right parentheis (−3,323) left parenthesis negative 3 comma 323 right parentheis (−6,8)

Answer 1

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

Let's evaluate the equation with each x-coordinate provided.

1. For the point \((-3, -55)\): \[ y = -6(-3)^2 + 3(-3) + 8 = -6(9) - 9 + 8 = -54 - 9 + 8 = -55 \] This point works.

2. For the point \((-6, 206)\): \[ y = -6(-6)^2 + 3(-6) + 8 = -6(36) - 18 + 8 = -216 - 18 + 8 = -226 \] This does not match.

3. For the point \((-3, 323)\): We already found that \((-3, -55)\) works, so we will not check this one since \(-55 \neq 323\).

4. For the point \((-6, 8)\): We found that for \(x = -6\), \(y\) evaluates to \(-226\), which is not 8.

The only point that satisfies the equation \( y = -6x^2 + 3x + 8 \) is \((-3, -55)\).

Thus, the answer is: \((-3, -55)\).

Answer 2

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 (−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. (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. (0.25,8.375) is the high point on the parabola. left parenthesis 0.25 comma 8.375 right parenthesis is the high point on the parabola.

Answer 3

To find the high or low point (vertex) of the parabola described by the quadratic equation \( y = -6x^2 + 3x + 8 \), we first note that this parabola opens downwards (since the coefficient of \(x^2\) is negative), which means it has a high point (maximum) rather than a low point (minimum).

1. Finding the Vertex: The x-coordinate of the vertex of a quadratic equation in the form \( y = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \] Here, \(a = -6\) and \(b = 3\): \[ x = -\frac{3}{2(-6)} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]

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

So, the vertex of the parabola is \((0.25, 8.375)\) and this point indicates the high point (maximum) of the parabola since it opens downward.

Now, let's evaluate the provided responses:

• (−1,−1) is the low point on the parabola.: Incorrect. This is neither high nor low.
• (0.5,9) is the high point on the parabola.: Incorrect. This point has not been calculated and doesn't represent the vertex.
• (0.25,8.375) is the low point on the parabola.: Incorrect. This point is the high point.
• (0.25,8.375) is the high point on the parabola.: Correct. This point is indeed the vertex and represents the maximum of the parabola.

Thus, the correct statement is: (0.25, 8.375) is the high point on the parabola.

Answer 4

Use the image to answer the question. Identify one of the zeros (or x-intercepts) on the graph. (1 point) Responses (0,2) left parenthesis 0 comma 2 right parenthesis x=1.5 x equals 1.5 (4,0) left parenthesis 4 comma 0 right parenthesis (0,−4) left parenthesis 0 comma negative 4 right parenthesis

Answer 5

To identify a zero (or x-intercept) on the graph, we typically look for points where the graph crosses or touches the x-axis. A zero of a function occurs where the value of \(y = 0\).

From the options provided:

1. (0, 2): This point is above the x-axis since \(y\) is positive. Not a zero.

2. \(x = 1.5\): This indicates a vertical line at \(x = 1.5\). We need to know the corresponding y-value at this x-coordinate to determine if it's a zero. Without the image, we cannot confirm if this point is a zero.

3. (4, 0): This point has \(y = 0\), which means it lies on the x-axis. This is indeed a zero.

4. (0, -4): This point is below the x-axis since \(y\) is negative. Not a zero.

Based on the analysis, the clear zero (or x-intercept) is:

(4, 0).

Answer 6

Use the image to answer the question. Identify the vertex on the graph. (1 point) Responses (−12,0) left parenthesis negative 12 comma 0 right parenthesis x=−11 x equals negative 11 (−10,0) left parenthesis negative 10 comma 0 right parenthesis (−11,−1) left parenthesis negative 11 comma negative 1 right parenthesis Skip to navigation

Answer 7

To identify the vertex of a parabola on a graph, we look for the highest or lowest point on the curve, depending on whether it opens upwards or downwards. The vertex can be represented in coordinate form \((x, y)\).

From the options provided:

1. (−12, 0): This point is on the x-axis and may not necessarily be the vertex unless the parabola is horizontal there.

2. \(x = -11\): This indicates a vertical line at \(x = -11\). Without additional context or a corresponding y-value, we cannot confirm if this represents the vertex.

3. (−10, 0): This point is also on the x-axis, and same as mentioned earlier, not necessarily the vertex.

4. (−11, −1): This point can represent a vertex if it is the lowest (minimum) or highest (maximum) point of the parabola.

Without the graph to visually inspect, the vertex is generally where the parabola changes direction. Typically, parabola vertices will not lie on the x-axis unless the parabola intersects there multiple times.

From these options, if the parabola opens upwards or downwards and (−11, −1) is the highest or lowest point, it would be the vertex.

Given the standard conventions in quadratic functions, the reasonable assumption would be:

(−11, −1) is likely the vertex of the parabola.

Answer 8

Use the image to answer the question. Determine which of the following statements is true about the parabola. (1 point) Responses The y-intercept and the x-intercepts share the same point. The y -intercept and the x -intercepts share the same point. The y-intercept and the vertex share the same point. The y -intercept and the vertex share the same point. The vertex and the x-intercepts share the same point. The vertex and the x -intercepts share the same point. There is only one x-intercept.

Answer 9

To evaluate which statement is true about the parabola based on the graph, let's clarify the concepts involved:

1. Y-intercept: The point where the graph intersects the y-axis (where \(x = 0\)).
2. X-intercepts: Points where the graph intersects the x-axis (where \(y = 0\)), which can be one or more points depending on the parabola.
3. Vertex: The highest or lowest point on the parabola, where it changes direction.

Now, let's analyze the statements one by one:

1. The y-intercept and the x-intercepts share the same point.: This would mean that the point where the parabola crosses the y-axis is also an x-intercept. This is unlikely unless it is a very specific case.

2. The y-intercept and the vertex share the same point.: This would mean that the point where the parabola crosses the y-axis is also the vertex. This can happen if the vertex lies on the y-axis, but it is generally not true for most parabolas.

3. The vertex and the x-intercepts share the same point.: This would mean that the vertex lies on the x-axis, making it an x-intercept, which is possible but not common.

4. There is only one x-intercept.: This can be true if the parabola is tangent to the x-axis (touches it at one point) or intersects at just one point but typically would be false if it intersects at two points.

Without seeing the graph, the last statement about having only one x-intercept tends to be a common arrangement when discussing parabolas. If the parabola opens either upwards or downwards and is shaped in such a way that it does not cross the x-axis more than once, then this statement would hold.

Thus, assuming the context of standard parabolic behavior, a reasonable answer could be:

There is only one x-intercept.

However, without the graph, we cannot ascertain the truth of the statements definitively. If the graph shows evident intersections, you may choose a different statement based on visible characteristics.