A Parabola, With Its Vertex At The Origin, Has A Directrix At $y=3$.Which Statements About The Parabola Are True? Select Two Options.A. The Focus Is Located At $(0,-3$\].B. The Parabola Opens To The Left.C. The $p$ Value Can

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Understanding the Basics of a Parabola

A parabola is a type of quadratic equation that can be represented in various forms, including the standard form $y = ax^2 + bx + c$. However, when a parabola has its vertex at the origin, it can be represented in the form $y = ax^2$, where $a$ is the coefficient that determines the direction and width of the parabola. In this case, we are dealing with a parabola that has its vertex at the origin and a directrix at $y = 3$. This means that the parabola opens either upwards or downwards, and the directrix is a horizontal line that is 3 units above the vertex.

The Focus of a Parabola

The focus of a parabola is a fixed point that is equidistant from the vertex and the directrix. In other words, it is the point where the parabola intersects a line that is perpendicular to the directrix and passes through the vertex. The focus is an important concept in the study of parabolas, as it helps to determine the shape and size of the parabola. In this case, since the directrix is at $y = 3$, the focus must be located at a point that is equidistant from the vertex and the directrix.

The p Value of a Parabola

The $p$ value of a parabola is the distance between the vertex and the focus. It is a measure of the width of the parabola and is used to determine the shape and size of the parabola. In this case, since the directrix is at $y = 3$, the $p$ value can be calculated as the distance between the vertex and the directrix, which is 3 units.

Determining the Focus and p Value

Based on the given information, we can determine the focus and $p$ value of the parabola. Since the directrix is at $y = 3$, the focus must be located at a point that is equidistant from the vertex and the directrix. This means that the focus is located at $(0, -3)$, which is 3 units below the vertex. Therefore, option A is correct.

Determining the Direction of the Parabola

To determine the direction of the parabola, we need to consider the sign of the coefficient $a$. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards. However, since the directrix is at $y = 3$, the parabola must open upwards. Therefore, option B is incorrect.

Conclusion

In conclusion, the focus of the parabola is located at $(0, -3)$, and the $p$ value is 3 units. The parabola opens upwards, and the directrix is at $y = 3$. Therefore, the correct statements about the parabola are:

• The focus is located at $(0, -3)$.
• The $p$ value is 3 units.

Frequently Asked Questions

• What is the vertex of a parabola? The vertex of a parabola is the point where the parabola intersects the axis of symmetry. In this case, the vertex is at the origin, which is $(0, 0)$.
• What is the directrix of a parabola? The directrix of a parabola is a horizontal line that is equidistant from the vertex and the focus. In this case, the directrix is at $y = 3$.
• What is the focus of a parabola? The focus of a parabola is a fixed point that is equidistant from the vertex and the directrix. In this case, the focus is located at $(0, -3)$.

References

• "Parabolas" by Math Open Reference. Retrieved from https://www.mathopenref.com/parabola.html
• "Directrix of a Parabola" by Purplemath. Retrieved from https://www.purplemath.com/modules/parabola/directrix.htm
• "Focus of a Parabola" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/parabolas-focus.html
  

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Understanding the Basics of a Parabola

A parabola is a type of quadratic equation that can be represented in various forms, including the standard form $y = ax^2 + bx + c$. However, when a parabola has its vertex at the origin, it can be represented in the form $y = ax^2$, where $a$ is the coefficient that determines the direction and width of the parabola. In this case, we are dealing with a parabola that has its vertex at the origin and a directrix at $y = 3$. This means that the parabola opens either upwards or downwards, and the directrix is a horizontal line that is 3 units above the vertex.

Q&A: Parabola with Vertex at the Origin and Directrix at $y=3$

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola intersects the axis of symmetry. In this case, the vertex is at the origin, which is $(0, 0)$.

Q: What is the directrix of a parabola?

A: The directrix of a parabola is a horizontal line that is equidistant from the vertex and the focus. In this case, the directrix is at $y = 3$.

Q: What is the focus of a parabola?

A: The focus of a parabola is a fixed point that is equidistant from the vertex and the directrix. In this case, the focus is located at $(0, -3)$.

Q: What is the p value of a parabola?

A: The $p$ value of a parabola is the distance between the vertex and the focus. It is a measure of the width of the parabola and is used to determine the shape and size of the parabola. In this case, the $p$ value is 3 units.

Q: How do I determine the direction of the parabola?

A: To determine the direction of the parabola, you need to consider the sign of the coefficient $a$. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards. However, since the directrix is at $y = 3$, the parabola must open upwards.

Q: What is the equation of a parabola with its vertex at the origin and a directrix at $y=3$?

A: The equation of a parabola with its vertex at the origin and a directrix at $y=3$ is $y = ax^2$, where $a$ is the coefficient that determines the direction and width of the parabola.

Q: How do I find the focus of a parabola?

A: To find the focus of a parabola, you need to use the formula $F = (h, k + p)$, where $(h, k)$ is the vertex of the parabola and $p$ is the distance between the vertex and the focus.

Q: What is the significance of the directrix of a parabola?

A: The directrix of a parabola is a horizontal line that is equidistant from the vertex and the focus. It is used to determine the shape and size of the parabola.

Q: How do I determine the p value of a parabola?

A: To determine the $p$ value of a parabola, you need to use the formula $p = \frac{1}{4a}$, where $a$ is the coefficient of the parabola.

Q: What is the relationship between the focus and the directrix of a parabola?

A: The focus and the directrix of a parabola are equidistant from the vertex. This means that the focus is located at a point that is the same distance from the vertex as the directrix.

Q: How do I graph a parabola with its vertex at the origin and a directrix at $y=3$?

A: To graph a parabola with its vertex at the origin and a directrix at $y=3$, you need to use the equation $y = ax^2$ and plot the points on a coordinate plane.

Q: What is the significance of the p value of a parabola?

A: The $p$ value of a parabola is a measure of the width of the parabola and is used to determine the shape and size of the parabola.

Q: How do I find the equation of a parabola with its vertex at the origin and a directrix at $y=3$?

A: To find the equation of a parabola with its vertex at the origin and a directrix at $y=3$, you need to use the formula $y = ax^2$, where $a$ is the coefficient that determines the direction and width of the parabola.

Q: What is the relationship between the vertex and the focus of a parabola?

A: The vertex and the focus of a parabola are equidistant from the directrix. This means that the vertex is located at a point that is the same distance from the directrix as the focus.

Q: How do I determine the equation of a parabola with its vertex at the origin and a directrix at $y=3$?

A: To determine the equation of a parabola with its vertex at the origin and a directrix at $y=3$, you need to use the formula $y = ax^2$, where $a$ is the coefficient that determines the direction and width of the parabola.

Q: What is the significance of the directrix of a parabola in real-world applications?

A: The directrix of a parabola is used in various real-world applications, such as the design of satellite dishes, telescopes, and other optical instruments.

Q: How do I find the focus of a parabola with its vertex at the origin and a directrix at $y=3$?

A: To find the focus of a parabola with its vertex at the origin and a directrix at $y=3$, you need to use the formula $F = (h, k + p)$, where $(h, k)$ is the vertex of the parabola and $p$ is the distance between the vertex and the focus.

Q: What is the relationship between the p value and the focus of a parabola?

A: The $p$ value and the focus of a parabola are related in that the $p$ value is the distance between the vertex and the focus.

Q: How do I determine the p value of a parabola with its vertex at the origin and a directrix at $y=3$?

A: To determine the $p$ value of a parabola with its vertex at the origin and a directrix at $y=3$, you need to use the formula $p = \frac{1}{4a}$, where $a$ is the coefficient of the parabola.

Q: What is the significance of the vertex of a parabola in real-world applications?

A: The vertex of a parabola is used in various real-world applications, such as the design of satellite dishes, telescopes, and other optical instruments.

Q: How do I graph a parabola with its vertex at the origin and a directrix at $y=3$?

A: To graph a parabola with its vertex at the origin and a directrix at $y=3$, you need to use the equation $y = ax^2$ and plot the points on a coordinate plane.

Q: What is the relationship between the vertex and the directrix of a parabola?

A: The vertex and the directrix of a parabola are equidistant from the focus. This means that the vertex is located at a point that is the same distance from the focus as the directrix.

Q: How do I find the equation of a parabola with its vertex at the origin and a directrix at $y=3$?

A: To find the equation of a parabola with its vertex at the origin and a directrix at $y=3$, you need to use the formula $y = ax^2$, where $a$ is the coefficient that determines the direction and width of the parabola.

Q: What is the significance of the p value of a parabola in real-world applications?

A: The $p$ value of a parabola is used in various real-world applications, such as the design of satellite dishes, telescopes, and other optical instruments.

Q: How do I determine the equation of a parabola with its vertex at the origin and a directrix at $y=3$?

A: To determine the equation of a parabola with its vertex at the origin and a directrix at $y=3$, you need to use the formula $y = ax^2$, where $a$ is the coefficient that determines the direction and width of the parabola.

Q: What is the relationship between the focus and the directrix of a parabola?

A: The focus and the directrix of a parabola are equidistant from the vertex. This means that the focus is located at a point that is the same distance from the vertex as the directrix.

Q: How do I find the focus of a parab