Which Statement Describes The Parabola $y = -x^2 - 4x - 26$?A. Vertex At (-2, -22), Directrix $y = -21.75$, Opens Down B. Vertex At (-2, -22), Directrix $y = -21.75$, Opens Up C. Vertex At (-2, 22), Directrix $y =

Introduction

Parabolas are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and computer science. A parabola is a type of quadratic equation that can be represented in the form of $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this article, we will focus on analyzing the parabola $y = -x^2 - 4x - 26$ and determine which statement describes it accurately.

The General Form of a Parabola

The general form of a parabola is given by the equation $y = ax^2 + bx + c$. The coefficient $a$ determines the direction in which the parabola opens. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards. The vertex of the parabola is the point where the parabola changes direction, and it can be found using the formula $x = -\frac{b}{2a}$.

Finding the Vertex of the Parabola

To find the vertex of the parabola $y = -x^2 - 4x - 26$, we need to use the formula $x = -\frac{b}{2a}$. In this case, $a = -1$ and $b = -4$. Plugging these values into the formula, we get:

$x = -\frac{-4}{2(-1)}$ $x = -\frac{-4}{-2}$ $x = 2$

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging the x-coordinate into the equation of the parabola:

$y = -(2)^2 - 4(2) - 26$ $y = -4 - 8 - 26$ $y = -38$

However, we can simplify this process by completing the square. To do this, we need to rewrite the equation of the parabola in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Completing the Square

To complete the square, we need to rewrite the equation of the parabola as follows:

$y = -x^2 - 4x - 26$ $y = -(x^2 + 4x) - 26$ $y = -(x^2 + 4x + 4) + 4 - 26$ $y = -(x + 2)^2 - 22$

Now that we have rewritten the equation of the parabola, we can see that the vertex is at $(-2, -22)$.

Determining the Directrix

The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola and does not touch the parabola. The equation of the directrix can be found using the formula $y = k - \frac{1}{4a}$, where $(h, k)$ is the vertex of the parabola.

In this case, the vertex is at $(-2, -22)$, and $a = -1$. Plugging these values into the formula, we get:

$y = -22 - \frac{1}{4(-1)}$ $y = -22 + \frac{1}{4}$ $y = -22 + 0.25$ $y = -21.75$

Conclusion

In conclusion, the parabola $y = -x^2 - 4x - 26$ has a vertex at $(-2, -22)$ and a directrix at $y = -21.75$. Since the parabola opens downwards, the correct statement is:

A. Vertex at (-2, -22), directrix $y = -21.75$, opens down

This statement accurately describes the parabola $y = -x^2 - 4x - 26$.

Introduction

In our previous article, we explored the concept of parabolas and analyzed the parabola $y = -x^2 - 4x - 26$. In this article, we will answer some frequently asked questions about parabolas and provide a comprehensive understanding of this mathematical concept.

Q: What is a parabola?

A: A parabola is a type of quadratic equation that can be represented in the form of $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The parabola is a U-shaped curve that can open upwards or downwards, depending on the value of $a$.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It can be found using the formula $x = -\frac{b}{2a}$, and the y-coordinate can be found by plugging the x-coordinate into the equation of the parabola.

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

A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$. Alternatively, you can complete the square to rewrite the equation of the parabola in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

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 and does not touch the parabola. The equation of the directrix can be found using the formula $y = k - \frac{1}{4a}$, where $(h, k)$ is the vertex of the parabola.

Q: How do I determine the direction in which a parabola opens?

A: To determine the direction in which a parabola opens, you need to look at the value of $a$. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards.

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

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

Q: How do I graph a parabola?

A: To graph a parabola, you can use the following steps:

1. Find the vertex of the parabola using the formula $x = -\frac{b}{2a}$.
2. Find the y-coordinate of the vertex by plugging the x-coordinate into the equation of the parabola.
3. Determine the direction in which the parabola opens.
4. Plot the vertex and the directrix on a coordinate plane.
5. Use the axis of symmetry to draw the parabola.

Q: What are some real-world applications of parabolas?

A: Parabolas have numerous real-world applications, including:

1. Physics: Parabolas are used to describe the trajectory of projectiles, such as thrown balls or rockets.
2. Engineering: Parabolas are used in the design of mirrors, lenses, and other optical systems.
3. Computer Science: Parabolas are used in computer graphics to create smooth curves and surfaces.
4. Architecture: Parabolas are used in the design of buildings and bridges to create aesthetically pleasing and functional structures.

Conclusion

In conclusion, parabolas are a fundamental concept in mathematics that have numerous real-world applications. By understanding the basics of parabolas, you can solve problems and create solutions in a variety of fields. We hope that this Q&A article has provided you with a comprehensive understanding of parabolas and has answered any questions you may have had.