Choose All Of The Equations Which Represent The Parabola With Vertex $(-4,-6$\] And Directrix $y=-2$.A. $(x+4)^2=-8(y+4$\]B. $(x+4)^2=-16(y+6$\]C. $(x+6)^2=-16(y+4$\]D. $8y=-x^2-8x-48$E.

Understanding the Basics of Parabolas

A parabola is a type of quadratic equation that can be represented in various forms. It is a U-shaped curve that can open upwards or downwards. The vertex of a parabola is the point where it changes direction, and the directrix is a line that is perpendicular to the axis of symmetry of the parabola. In this article, we will focus on choosing the correct equation that represents a parabola with a given vertex and directrix.

Vertex and Directrix

The vertex of the parabola is given as $(-4,-6)$, which means that the parabola opens upwards or downwards with its vertex at the point $(-4,-6)$. The directrix is given as $y=-2$, which is a horizontal line that is 2 units above the vertex.

Equation of a Parabola

The general equation of a parabola with vertex $(h,k)$ and directrix $y=d$ is given by:

$$\frac{(x-h)^2}{4p} = -(y-k)$$

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

Choosing the Correct Equation

We are given five options to choose from:

A. $(x+4)^2=-8(y+4)$ B. $(x+4)^2=-16(y+6)$ C. $(x+6)^2=-16(y+4)$ D. $8y=-x^2-8x-48$ E. $(x+4)^2=-8(y+6)$

To choose the correct equation, we need to analyze each option and compare it with the general equation of a parabola.

Option A

$(x+4)^2=-8(y+4)$

This equation can be rewritten as:

$$\frac{(x+4)^2}{-8} = -(y+4)$$

Comparing this with the general equation, we can see that $h=-4$, $k=-4$, and $p=1$. However, the distance between the vertex and the directrix is 2 units, which is not equal to $p$. Therefore, this equation is not correct.

Option B

$(x+4)^2=-16(y+6)$

This equation can be rewritten as:

$$\frac{(x+4)^2}{-16} = -(y+6)$$

Comparing this with the general equation, we can see that $h=-4$, $k=-6$, and $p=1$. However, the distance between the vertex and the directrix is 2 units, which is not equal to $p$. Therefore, this equation is not correct.

Option C

$(x+6)^2=-16(y+4)$

This equation can be rewritten as:

$$\frac{(x+6)^2}{-16} = -(y+4)$$

Comparing this with the general equation, we can see that $h=-6$, $k=-4$, and $p=1$. However, the distance between the vertex and the directrix is 2 units, which is not equal to $p$. Therefore, this equation is not correct.

Option D

$8y=-x^2-8x-48$

This equation can be rewritten as:

$$y = -\frac{x^2+8x+48}{8}$$

Comparing this with the general equation, we can see that $h=-4$, $k=-6$, and $p=2$. The distance between the vertex and the directrix is 2 units, which is equal to $p$. Therefore, this equation is correct.

Option E

$(x+4)^2=-8(y+6)$

This equation can be rewritten as:

$$\frac{(x+4)^2}{-8} = -(y+6)$$

Comparing this with the general equation, we can see that $h=-4$, $k=-6$, and $p=1$. However, the distance between the vertex and the directrix is 2 units, which is not equal to $p$. Therefore, this equation is not correct.

Conclusion

In conclusion, the correct equation that represents the parabola with vertex $(-4,-6)$ and directrix $y=-2$ is:

$8y=-x^2-8x-48$

This equation satisfies the conditions of the problem, and it is the only option that does so.

References

• [1] "Parabola" by Math Open Reference. Retrieved 2023-02-20.
• [2] "Equation of a Parabola" by Purplemath. Retrieved 2023-02-20.

Note

Q: What is a parabola?

A: A parabola is a type of quadratic equation that can be represented in various forms. It is a U-shaped curve that can open upwards or downwards.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where it changes direction. It is the lowest or highest point on the curve.

Q: What is the directrix of a parabola?

A: The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola. It is a horizontal line that is a certain distance away from the vertex.

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

A: To find the equation of a parabola, you need to know the coordinates of the vertex and the directrix. You can use the general equation of a parabola to find the equation:

$$\frac{(x-h)^2}{4p} = -(y-k)$$

where $h$ and $k$ are the coordinates of the vertex, and $p$ is the distance between the vertex and the directrix.

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

A: The axis of symmetry of a parabola is a vertical line that passes through the vertex. It is a line that divides the parabola into two equal parts.

Q: How do I graph a parabola?

A: To graph a parabola, you need to know the equation of the parabola. You can use a graphing calculator or a computer program to graph the parabola. You can also use a piece of graph paper and a pencil to draw the parabola by hand.

Q: What is the focus of a parabola?

A: The focus of a parabola is a point that is a certain distance away from the vertex. It is a point that is on the axis of symmetry of the parabola.

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

A: To find the focus of a parabola, you need to know the equation of the parabola. You can use the general equation of a parabola to find the focus:

$$F = (h, k+p)$$

where $h$ and $k$ are the coordinates of the vertex, and $p$ is the distance between the vertex and the directrix.

Q: What is the latus rectum of a parabola?

A: The latus rectum of a parabola is a line that passes through the focus and is perpendicular to the axis of symmetry of the parabola. It is a line that is a certain distance away from the vertex.

Q: How do I find the latus rectum of a parabola?

A: To find the latus rectum of a parabola, you need to know the equation of the parabola. You can use the general equation of a parabola to find the latus rectum:

$$LR = \frac{4p}{a}$$

where $p$ is the distance between the vertex and the directrix, and $a$ is the coefficient of the $x^2$ term in the equation of the parabola.

Q: What is the standard form of a parabola?

A: The standard form of a parabola is:

$$y = a(x-h)^2 + k$$

where $a$, $h$, and $k$ are constants.

Q: How do I convert a parabola from standard form to vertex form?

A: To convert a parabola from standard form to vertex form, you need to complete the square on the $x$ term. You can do this by adding and subtracting the square of half the coefficient of the $x$ term.

Q: How do I convert a parabola from vertex form to standard form?

A: To convert a parabola from vertex form to standard form, you need to expand the squared term. You can do this by multiplying out the squared term and combining like terms.

Q: What is the difference between a parabola and a hyperbola?

A: A parabola is a U-shaped curve that can open upwards or downwards, while a hyperbola is a curve that has two branches that open in opposite directions.