Given The Directrix $x = 6$ And The Focus $(3, -5$\], What Is The Vertex Form Of The Equation Of The Parabola?The Vertex Form Of The Equation Is $x = \square (y + \square)^2 + \square$. Type The Correct Answer In Each Box. Use

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Introduction

In mathematics, a parabola is a quadratic curve that can be represented in various forms, including the standard form, vertex form, and focus-directrix form. The vertex form of a parabola is a powerful tool for analyzing and graphing parabolas, as it provides a clear and concise representation of the parabola's shape and position. In this article, we will explore the vertex form of a parabola and provide a step-by-step guide on how to find the vertex form of a parabola given its directrix and focus.

Understanding the Vertex Form

The vertex form of a parabola is given by the equation:

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

where (h, k) is the vertex of the parabola, and a is a constant that determines the parabola's shape and direction. The vertex form is a useful tool for analyzing and graphing parabolas, as it provides a clear and concise representation of the parabola's shape and position.

Finding the Vertex Form of a Parabola

To find the vertex form of a parabola, we need to know the directrix and focus of the parabola. The directrix is a line that is perpendicular to the axis of symmetry of the parabola, and the focus is a point that is equidistant from the directrix and the vertex of the parabola.

Given the directrix x = 6 and the focus (3, -5), we can use the following steps to find the vertex form of the equation of the parabola:

Step 1: Determine the Axis of Symmetry

The axis of symmetry of a parabola is a line that passes through the vertex of the parabola and is perpendicular to the directrix. In this case, the directrix is x = 6, so the axis of symmetry is a vertical line that passes through x = 3.

Step 2: Find the Vertex of the Parabola

The vertex of a parabola is the point that is equidistant from the directrix and the focus. Since the directrix is x = 6 and the focus is (3, -5), the vertex of the parabola is the point (3, -5).

Step 3: Determine the Value of a

The value of a in the vertex form of a parabola determines the parabola's shape and direction. To find the value of a, we need to use the fact that the parabola passes through the focus (3, -5). Substituting the values of x and y into the vertex form of the equation, we get:

3 = a(-5 - (-5))^2 + 3

Simplifying the equation, we get:

3 = a(0)^2 + 3

Since (0)^2 = 0, the equation becomes:

3 = a(0) + 3

This equation is true for any value of a, so we can conclude that a = 1.

Step 4: Write the Vertex Form of the Equation

Now that we have found the value of a, we can write the vertex form of the equation of the parabola:

x = 1(y - (-5))^2 + 3

Simplifying the equation, we get:

x = 1(y + 5)^2 + 3

Conclusion

In this article, we have explored the vertex form of a parabola and provided a step-by-step guide on how to find the vertex form of a parabola given its directrix and focus. We have used the directrix x = 6 and the focus (3, -5) to find the vertex form of the equation of the parabola, which is x = 1(y + 5)^2 + 3. We hope that this article has provided a useful tool for analyzing and graphing parabolas, and that it has helped to clarify the concept of the vertex form of a parabola.

Key Takeaways

  • The vertex form of a parabola is a powerful tool for analyzing and graphing parabolas.
  • The vertex form of a parabola is given by the equation x = a(y - k)^2 + h, where (h, k) is the vertex of the parabola, and a is a constant that determines the parabola's shape and direction.
  • To find the vertex form of a parabola, we need to know the directrix and focus of the parabola.
  • The directrix is a line that is perpendicular to the axis of symmetry of the parabola, and the focus is a point that is equidistant from the directrix and the vertex of the parabola.
  • The value of a in the vertex form of a parabola determines the parabola's shape and direction.
  • The vertex form of the equation of a parabola can be written as x = a(y - k)^2 + h, where (h, k) is the vertex of the parabola, and a is a constant that determines the parabola's shape and direction.
    Parabola Vertex Form: A Q&A Guide =====================================

Introduction

In our previous article, we explored the vertex form of a parabola and provided a step-by-step guide on how to find the vertex form of a parabola given its directrix and focus. In this article, we will answer some of the most frequently asked questions about the vertex form of a parabola.

Q&A

Q: What is the vertex form of a parabola?

A: The vertex form of a parabola is a powerful tool for analyzing and graphing parabolas. It is given by the equation x = a(y - k)^2 + h, where (h, k) is the vertex of the parabola, and a is a constant that determines the parabola's shape and direction.

Q: How do I find the vertex form of a parabola given its directrix and focus?

A: To find the vertex form of a parabola, you need to know the directrix and focus of the parabola. The directrix is a line that is perpendicular to the axis of symmetry of the parabola, and the focus is a point that is equidistant from the directrix and the vertex of the parabola. You can use the following steps to find the vertex form of the equation of the parabola:

  1. Determine the axis of symmetry of the parabola.
  2. Find the vertex of the parabola.
  3. Determine the value of a in the vertex form of the equation.
  4. Write the vertex form of the equation.

Q: What is the significance of the value of a in the vertex form of a parabola?

A: The value of a in the vertex form of a parabola determines the parabola's shape and direction. If a is positive, the parabola opens upward. If a is negative, the parabola opens downward.

Q: How do I determine the value of a in the vertex form of a parabola?

A: To determine the value of a in the vertex form of a parabola, you need to use the fact that the parabola passes through the focus. You can substitute the values of x and y into the vertex form of the equation and solve for a.

Q: What is the vertex form of the equation of a parabola with a horizontal axis of symmetry?

A: The vertex form of the equation of a parabola with a horizontal axis of symmetry is given by the equation y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola, and a is a constant that determines the parabola's shape and direction.

Q: How do I graph a parabola in vertex form?

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

  1. Determine the vertex of the parabola.
  2. Determine the value of a in the vertex form of the equation.
  3. Plot the vertex of the parabola on the coordinate plane.
  4. Plot two points on either side of the vertex that are equidistant from the vertex.
  5. Draw a smooth curve through the points to form the parabola.

Conclusion

In this article, we have answered some of the most frequently asked questions about the vertex form of a parabola. We hope that this article has provided a useful tool for analyzing and graphing parabolas, and that it has helped to clarify the concept of the vertex form of a parabola.

Key Takeaways

  • The vertex form of a parabola is a powerful tool for analyzing and graphing parabolas.
  • The vertex form of a parabola is given by the equation x = a(y - k)^2 + h, where (h, k) is the vertex of the parabola, and a is a constant that determines the parabola's shape and direction.
  • To find the vertex form of a parabola, you need to know the directrix and focus of the parabola.
  • The value of a in the vertex form of a parabola determines the parabola's shape and direction.
  • The vertex form of the equation of a parabola can be written as x = a(y - k)^2 + h, where (h, k) is the vertex of the parabola, and a is a constant that determines the parabola's shape and direction.