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.

Question

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.

(−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.

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

(0.5,9) is the high point on the parabola.

Answer 1

To determine the high or low point of the parabola represented by the equation \( y = -6x^2 + 3x + 8 \), we first need to calculate the vertex of the parabola, since the vertex will give us either the highest or lowest point, depending on the orientation of the parabola.

The formula for the x-coordinate of the vertex of a quadratic function in the form \( y = ax^2 + bx + c \) is given by:

\[ x = -\frac{b}{2a} \]

In this case:

• \( a = -6 \)
• \( b = 3 \)

Substituting in these values:

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

Next, we substitute \( x = 0.25 \) back into the quadratic equation to find the corresponding \( y \)-coordinate:

\[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ y = -6(0.0625) + 0.75 + 8 \] \[ y = -0.375 + 0.75 + 8 \] \[ y = 8.375 \]

So, the vertex of the parabola is \( (0.25, 8.375) \).

Since the coefficient of \( x^2 \) (which is -6) is negative, the parabola opens downward, meaning that the vertex represents a high point.

Thus, the correct option is:

(0.25, 8.375) is the high point on the parabola.

Answer 2

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

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

x=1.5
x equals 1.5

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

(4,0)

Answer 3

To identify one of the zeros (or x-intercepts) of the parabola from the provided options, we need to determine which of the given points is where the parabola intersects the x-axis.

The x-intercepts are points where \( y = 0 \). Therefore, we are looking for an option where the y-coordinate is 0.

• (0, 2): This point is not an x-intercept because the y-coordinate is 2.
• x = 1.5: This is just a value of x and does not specify a point with y = 0.
• (0, -4): This point is not an x-intercept because the y-coordinate is -4.
• (4, 0): This point is an x-intercept since the y-coordinate is 0.

Thus, the correct response is:

(4, 0).

Answer 4

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

(−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)

Answer 5

To identify the vertex of the parabola from the provided options, we need to determine which of the given coordinates represents the vertex, typically characterized as the highest or lowest point of the parabola.

Here are the options and analysis:

• (−12, 0): This point could be a location on the parabola, but it is not specified whether it is a vertex without additional context about the shape of the graph.
• x = −11: This option does not provide a full coordinate (it gives only the x-coordinate and does not indicate a y-coordinate).
• (−10, 0): Similar to the first option, this point could be on the parabola but does not specify that it is the vertex.
• (−11, −1): This indicates a point where the x-coordinate is -11 and the y-coordinate is -1. This could likely be a vertex if it is the lowest or highest point.

Based on these interpretations, we would expect the vertex to likely have the form of a point where either it appears to be the lowest or highest y-value in the given x-range.

If the parabola opens upwards, then the lowest point would represent the vertex, and if it opens downwards, then the highest point would represent the vertex.

Assuming typical characteristics of a vertex: Without the visual representation of the graph, we would most reasonably choose (−11, −1), as it gives us both coordinates and is likely the vertex based on the potential shape of a parabola.

Thus, the correct response is:

(−11, −1).

Answer 6

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

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

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

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 x-intercepts share the same point.

Answer 7

To determine which statement about the parabola is true, we can analyze the nature of the vertex, y-intercept, and x-intercepts.

Definitions:

1. Vertex: The highest or lowest point of the parabola, depending on whether it opens upwards or downwards.
2. Y-Intercept: The point where the parabola intersects the y-axis. This occurs when \( x = 0 \).
3. X-Intercept(s): The point(s) where the parabola intersects the x-axis, which occurs when \( y = 0 \).

Analyzing Each Statement:

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

   • This statement is true only if the point where the parabola intersects the y-axis is also the vertex. This is rare but possible in certain parabolas.

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

   • This statement is true if the parabola only touches the x-axis at one point, characteristic of a perfect square.

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

   • This means that the vertex lies on the x-axis. This can happen if the vertex is the minimum point of an upwards-opening parabola (the vertex is at the tip of the parabola just touching the x-axis). It's generally not too common for a parabola to have its vertex as an x-intercept.

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

   • This is the case when the point at which the parabola intersects the y-axis (at \( x = 0 \)) is also where it intersects the x-axis (at \( y = 0 \)). Usually, this does not occur unless the parabola is degenerate (not a standard case).

Correct Choice:

Without seeing the specific graph, it is difficult to determine the precise relationships between these points. However, typically for a standard parabola, the most likely true statement among the responses would be:

• "There is only one x-intercept."

This can happen in cases where the parabola just touches the x-axis.

If the graph clearly demonstrates a different relationship (for instance, the vertex being at the y-intercept), you may need to revisit the visualization to confirm, but this statement has a higher likelihood of being true in general cases.