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

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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, vertex form, and focus-directrix form. In this article, we will focus on a parabola with its vertex at the origin and a directrix at $y = 3$. This information will help us determine the characteristics of the parabola, including its focus, direction of opening, and the value of $p$.

The Focus-Directrix Form of a Parabola

The focus-directrix form of a parabola is given by the equation:

$$\frac{(x - h)^2}{p^2} = \frac{(y - k)^2}{p^2}$$

where $(h, k)$ is the vertex of the parabola, and $p$ is the distance between the vertex and the focus. The directrix of the parabola is a line that is perpendicular to the axis of symmetry and is located at a distance $p$ from the vertex.

The Vertex at the Origin

Since the vertex of the parabola is at the origin, we can write the equation of the parabola as:

$$\frac{x^2}{p^2} = \frac{y^2}{p^2}$$

The Directrix at $y = 3$

The directrix of the parabola is at $y = 3$, which means that the parabola opens either upwards or downwards. Since the vertex is at the origin, the parabola must open upwards.

The Focus of the Parabola

The focus of the parabola is located at a distance $p$ from the vertex, and since the parabola opens upwards, the focus must be located above the vertex. Therefore, the focus is located at $(0, p)$.

The Value of $p$

Since the directrix is at $y = 3$, the distance between the vertex and the directrix is $3$. Therefore, the value of $p$ is $3$.

The Parabola Opens Upwards

Since the directrix is at $y = 3$, the parabola opens upwards. This means that the parabola is symmetric about the $x$-axis and opens in the positive $y$-direction.

The Focus is Located at $(0, 3)$

Since the value of $p$ is $3$, the focus is located at $(0, 3)$.

Conclusion

In conclusion, the statements about the parabola that are true are:

• The parabola opens upwards.
• The focus is located at $(0, 3)$.

The other options are not true. The parabola does not open to the left, and the value of $p$ is $3$, not $-3$.

Final Answer

The final answer is:

• The parabola opens upwards.
• The focus is located at $(0, 3)$.

Note: The final answer is not a numerical value, but rather a statement about the characteristics of the parabola.

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Understanding the Basics of a Parabola

In our previous article, we discussed the characteristics of a parabola with its vertex at the origin and a directrix at $y = 3$. In this article, we will answer some frequently asked questions about this type of parabola.

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 given by:

$$\frac{x^2}{p^2} = \frac{y^2}{p^2}$$

where $p$ is the distance between the vertex and the focus.

Q: What is the value of $p$ for this parabola?

A: The value of $p$ for this parabola is $3$, since the directrix is at $y = 3$.

Q: Where is the focus of the parabola located?

A: The focus of the parabola is located at $(0, 3)$, since the value of $p$ is $3$.

Q: Does the parabola open to the left or to the right?

A: The parabola opens upwards, since the directrix is at $y = 3$. This means that the parabola is symmetric about the $x$-axis and opens in the positive $y$-direction.

Q: What is the axis of symmetry of the parabola?

A: The axis of symmetry of the parabola is the $x$-axis, since the parabola is symmetric about the $x$-axis.

Q: What is the vertex of the parabola?

A: The vertex of the parabola is at the origin, $(0, 0)$.

Q: What is the directrix of the parabola?

A: The directrix of the parabola is the line $y = 3$.

Q: What is the distance between the vertex and the focus?

A: The distance between the vertex and the focus is $p = 3$.

Q: What is the distance between the vertex and the directrix?

A: The distance between the vertex and the directrix is also $p = 3$.

Q: Can the parabola be written in the standard form?

A: Yes, the parabola can be written in the standard form:

$$y^2 = 4px$$

where $p = 3$.

Q: What is the value of $4p$ for this parabola?

A: The value of $4p$ for this parabola is $12$, since $p = 3$.

Q: What is the equation of the directrix in the standard form?

A: The equation of the directrix in the standard form is $y = -p = -3$.

Q: What is the equation of the axis of symmetry in the standard form?

A: The equation of the axis of symmetry in the standard form is $x = 0$.

Q: What is the equation of the parabola in the vertex form?

A: The equation of the parabola in the vertex form is:

$$y^2 = 4(3)x$$

Q: What is the value of $4(3)$ for this parabola?

A: The value of $4(3)$ for this parabola is $12$.

Q: What is the equation of the directrix in the vertex form?

A: The equation of the directrix in the vertex form is $y = -3$.

Q: What is the equation of the axis of symmetry in the vertex form?

A: The equation of the axis of symmetry in the vertex form is $x = 0$.

Conclusion

In conclusion, we have answered some frequently asked questions about a parabola with its vertex at the origin and a directrix at $y = 3$. We hope that this article has been helpful in understanding the characteristics of this type of parabola.