Drag Each Sign Or Value To The Correct Location On The Equation. Each Sign Or Value Can Be Used More Than Once, But Not All Signs And Values Will Be Used.The Focus Of A Parabola Is { (-4,-5)$}$, And Its Directrix Is { Y=-1$}$. Fill

by ADMIN 232 views

Introduction

In mathematics, a parabola is a type of quadratic equation that has a specific focus and directrix. The focus of a parabola is a fixed point that is equidistant from the directrix and the parabola itself. In this article, we will explore how to solve a parabola equation using the given focus and directrix.

Understanding the Focus and Directrix

The focus of a parabola is a point that is equidistant from the directrix and the parabola itself. The directrix is a line that is perpendicular to the axis of symmetry of the parabola. In this case, the focus is given as {(-4,-5)$}$ and the directrix is given as {y=-1$}$.

The Equation of a Parabola

The equation of a parabola can be written in the form y=a(x−h)2+k{y = a(x-h)^2 + k}, where (h,k){(h,k)} is the vertex of the parabola. However, in this case, we are given the focus and directrix, so we can use the following equation to solve for the parabola:

(x−h)2=4p(y−k){(x-h)^2 = 4p(y-k)}

where (h,k){(h,k)} is the vertex of the parabola, and p{p} is the distance from the vertex to the focus.

Solving for the Parabola

To solve for the parabola, we need to find the values of h{h}, k{k}, and p{p}. We are given the focus {(-4,-5)$}$ and the directrix {y=-1$}$. We can use this information to find the values of h{h}, k{k}, and p{p}.

Step 1: Find the Value of p

The value of p{p} is the distance from the vertex to the focus. Since the focus is {(-4,-5)$}$ and the directrix is {y=-1$}$, we can find the value of p{p} by finding the distance from the vertex to the focus.

Step 2: Find the Value of h

The value of h{h} is the x-coordinate of the vertex. Since the focus is {(-4,-5)$}$, we can find the value of h{h} by finding the x-coordinate of the vertex.

Step 3: Find the Value of k

The value of k{k} is the y-coordinate of the vertex. Since the focus is {(-4,-5)$}$, we can find the value of k{k} by finding the y-coordinate of the vertex.

Step 4: Write the Equation of the Parabola

Once we have found the values of h{h}, k{k}, and p{p}, we can write the equation of the parabola using the following formula:

(x−h)2=4p(y−k){(x-h)^2 = 4p(y-k)}

Step 5: Solve for the Parabola

To solve for the parabola, we need to plug in the values of h{h}, k{k}, and p{p} into the equation and simplify.

The Final Answer

After solving for the parabola, we get the following equation:

(x+4)2=4(3)(y+5){(x+4)^2 = 4(3)(y+5)}

This is the equation of the parabola with the given focus and directrix.

Conclusion

In this article, we have explored how to solve a parabola equation using the given focus and directrix. We have found the values of h{h}, k{k}, and p{p} and written the equation of the parabola. We have also solved for the parabola and found the final answer.

Key Takeaways

  • The focus of a parabola is a point that is equidistant from the directrix and the parabola itself.
  • The directrix is a line that is perpendicular to the axis of symmetry of the parabola.
  • The equation of a parabola can be written in the form y=a(x−h)2+k{y = a(x-h)^2 + k}, where (h,k){(h,k)} is the vertex of the parabola.
  • The value of p{p} is the distance from the vertex to the focus.
  • The value of h{h} is the x-coordinate of the vertex.
  • The value of k{k} is the y-coordinate of the vertex.
  • The equation of the parabola can be written using the following formula: (x−h)2=4p(y−k){(x-h)^2 = 4p(y-k)}

Frequently Asked Questions

  • What is the focus of a parabola?
  • What is the directrix of a parabola?
  • How do I find the values of h{h}, k{k}, and p{p}?
  • How do I write the equation of the parabola?
  • How do I solve for the parabola?

References

Introduction

In our previous article, we explored how to solve a parabola equation using the given focus and directrix. However, we know that there are many more questions that you may have about parabolas. In this article, we will answer some of the most frequently asked questions about parabolas.

Q: What is the focus of a parabola?

A: The focus of a parabola is a point that is equidistant from the directrix and the parabola itself. It is a fixed point that is used to define 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. It is a line that is used to define the parabola.

Q: How do I find the values of h, k, and p?

A: To find the values of h, k, and p, you need to use the given focus and directrix. The value of p is the distance from the vertex to the focus, the value of h is the x-coordinate of the vertex, and the value of k is the y-coordinate of the vertex.

Q: How do I write the equation of the parabola?

A: To write the equation of the parabola, you need to use the following formula: (x-h)^2 = 4p(y-k). You need to plug in the values of h, k, and p into the equation and simplify.

Q: How do I solve for the parabola?

A: To solve for the parabola, you need to plug in the values of h, k, and p into the equation and simplify. You can then use the equation to find the x and y coordinates of the parabola.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is the point where the parabola is at its minimum or maximum value.

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

A: To find the vertex of a parabola, you need to use the given focus and directrix. The vertex is the point where the parabola is at its minimum or maximum value.

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. It is a line that is used to define the parabola.

Q: How do I find the axis of symmetry of a parabola?

A: To find the axis of symmetry of a parabola, you need to use the given focus and directrix. The axis of symmetry is a line that passes through the vertex of the parabola.

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

A: The equation of a parabola in vertex form is y = a(x-h)^2 + k, where (h,k) is the vertex of the parabola.

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

A: To convert the equation of a parabola from standard form to vertex form, you need to complete the square. You can then rewrite the equation in vertex form.

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

A: A parabola is a type of quadratic equation that has a specific focus and directrix. A circle is a type of quadratic equation that has a specific center and radius.

Q: How do I graph a parabola?

A: To graph a parabola, you need to use the given focus and directrix. You can then use the equation to find the x and y coordinates of the parabola.

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

A: Parabolas have many real-world applications, including the design of satellite dishes, the shape of a thrown ball, and the design of a parabolic mirror.

Conclusion

In this article, we have answered some of the most frequently asked questions about parabolas. We hope that this article has been helpful in understanding the concept of parabolas and how to solve parabola equations.

Key Takeaways

  • The focus of a parabola is a point that is equidistant from the directrix and the parabola itself.
  • The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola.
  • The equation of a parabola can be written in the form (x-h)^2 = 4p(y-k), where (h,k) is the vertex of the parabola.
  • The vertex of a parabola is the point where the parabola changes direction.
  • The axis of symmetry of a parabola is a line that passes through the vertex of the parabola.
  • The equation of a parabola in vertex form is y = a(x-h)^2 + k, where (h,k) is the vertex of the parabola.

References