Which Of The Following Statements Are True About The Graph Of { (y+5)^2=10(x-2)$}$? Choose Three Correct Answers.A. The Focus Is Found Using The Formula { (h, K+p)$}$.B. The Directrix Is At { X=-2.5$}$.C.

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Introduction

When dealing with the graph of a parabola, it's essential to understand the various components that make up its structure. The focus, directrix, and vertex are crucial elements that help us visualize and analyze the parabola's behavior. In this article, we will delve into the world of parabolas and explore the statements related to the graph of {(y+5)^2=10(x-2)$}$. We will examine each statement and determine its validity, providing a comprehensive understanding of the parabola's characteristics.

The Focus of a Parabola

The focus of a parabola is a point that lies on the axis of symmetry and is equidistant from the vertex and the directrix. The formula for finding the focus of a parabola is {(h, k+p)$}$, where (h,k){(h, k)} is the vertex and p{p} is the distance from the vertex to the focus. Let's examine statement A: "The focus is found using the formula {(h, k+p)$}{{content}}quot;.

Statement A: The Focus is Found Using the Formula {(h, k+p)$}$

The formula {(h, k+p)$}$ is indeed used to find the focus of a parabola. However, it's essential to note that this formula is applicable when the parabola is in the form {(y-k)^2=4p(x-h)$}$. In the given equation {(y+5)^2=10(x-2)$}$, the parabola is in the form {(y-k)^2=4ap(x-h)$}$, where a=5{a=5} and p=2.5{p=2.5}. Therefore, the focus of the parabola can be found using the formula {(h, k+p)$}$, making statement A a true statement.

The Directrix of a Parabola

The directrix of a parabola is a line that is perpendicular to the axis of symmetry and is located at a distance p{p} from the vertex. The equation of the directrix can be found using the formula x=h−p{x=h-p} or x=h+p{x=h+p}, depending on the orientation of the parabola. Let's examine statement B: "The directrix is at {x=-2.5$}{{content}}quot;.

Statement B: The Directrix is at {x=-2.5$}$

To determine the equation of the directrix, we need to find the value of p{p}. In the given equation {(y+5)^2=10(x-2)$}$, we can rewrite it in the form {(y-k)^2=4ap(x-h)$}$ as {(y+5)^2=20(x-2)$}$. Comparing this with the standard form, we can see that a=5{a=5} and p=2.5{p=2.5}. Since the parabola opens to the right, the directrix is located at x=h−p{x=h-p}. Substituting the values of h{h} and p{p}, we get x=2−2.5=−0.5{x=2-2.5=-0.5}. Therefore, statement B is incorrect, and the directrix is not at {x=-2.5$}$.

The Vertex of a Parabola

The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola, depending on its orientation. The vertex can be found using the formula (h,k){(h, k)}, where h{h} is the x-coordinate and k{k} is the y-coordinate. Let's examine statement C: "The vertex is at (2,−5)${(2, -5)\$}".

Statement C: The Vertex is at (2,−5)${(2, -5)\$}

To find the vertex, we need to rewrite the given equation in the form {(y-k)^2=4ap(x-h)$}$. We can rewrite the equation {(y+5)^2=10(x-2)$}$ as {(y+5)^2=20(x-2)$}$. Comparing this with the standard form, we can see that a=5{a=5} and p=2.5{p=2.5}. The vertex is located at (h,k){(h, k)}, where h{h} is the x-coordinate and k{k} is the y-coordinate. In this case, h=2{h=2} and k=−5{k=-5}. Therefore, statement C is correct, and the vertex is indeed at (2,−5)${(2, -5)\$}.

Conclusion

In conclusion, we have examined three statements related to the graph of {(y+5)^2=10(x-2)$}$. We found that statement A is true, statement B is false, and statement C is true. The focus of the parabola can be found using the formula {(h, k+p)$}$, the directrix is not at {x=-2.5$}$, and the vertex is indeed at (2,−5)${(2, -5)\$}. This comprehensive analysis has provided a deeper understanding of the parabola's characteristics and has helped us determine the validity of each statement.

Key Takeaways

  • The focus of a parabola can be found using the formula {(h, k+p)$}$.
  • The directrix of a parabola is a line that is perpendicular to the axis of symmetry and is located at a distance p{p} from the vertex.
  • The vertex of a parabola is the point where the parabola changes direction and is the minimum or maximum point of the parabola, depending on its orientation.

Final Thoughts

Understanding the graph of a parabola is crucial in mathematics, and it's essential to have a comprehensive knowledge of its characteristics. By analyzing the statements related to the graph of {(y+5)^2=10(x-2)$}$, we have gained a deeper understanding of the parabola's behavior and have determined the validity of each statement. This knowledge will help us in solving various mathematical problems and will provide a solid foundation for further studies in mathematics.

Introduction

In our previous article, we delved into the world of parabolas and explored the statements related to the graph of {(y+5)^2=10(x-2)$}$. We examined each statement and determined its validity, providing a comprehensive understanding of the parabola's characteristics. In this article, we will address some of the most frequently asked questions related to the graph of a parabola.

Q: What is the focus of a parabola?

A: The focus of a parabola is a point that lies on the axis of symmetry and is equidistant from the vertex and the directrix. It is a crucial element in understanding the parabola's behavior.

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

A: To find the focus of a parabola, you can use the formula {(h, k+p)$}$, where (h,k){(h, k)} is the vertex and p{p} is the distance from the vertex to the focus.

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 and is located at a distance p{p} from the vertex. It is a key element in understanding the parabola's behavior.

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

A: To find the directrix of a parabola, you can use the formula x=h−p{x=h-p} or x=h+p{x=h+p}, depending on the orientation 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 minimum or maximum point of the parabola, depending on its orientation.

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

A: To find the vertex of a parabola, you can use the formula (h,k){(h, k)}, where h{h} is the x-coordinate and k{k} is the y-coordinate.

Q: What is the equation of a parabola?

A: The equation of a parabola can be written in the form {(y-k)^2=4ap(x-h)$}$, where a{a} is the coefficient of the squared term, p{p} is the distance from the vertex to the focus, and (h,k){(h, k)} is the vertex.

Q: How do I graph a parabola?

A: To graph a parabola, you can start by finding the vertex and the focus. Then, use the equation of the parabola to find the x-intercepts and the y-intercepts. Finally, plot the points and draw 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 trajectory of a projectile.

Conclusion

In conclusion, we have addressed some of the most frequently asked questions related to the graph of a parabola. We have provided a comprehensive understanding of the parabola's characteristics and have offered tips and tricks for finding the focus, directrix, and vertex. Whether you are a student or a professional, understanding the graph of a parabola is essential in mathematics and has many real-world applications.

Key Takeaways

  • The focus of a parabola is a point that lies on the axis of symmetry and is equidistant from the vertex and the directrix.
  • The directrix of a parabola is a line that is perpendicular to the axis of symmetry and is located at a distance p{p} from the vertex.
  • The vertex of a parabola is the point where the parabola changes direction and is the minimum or maximum point of the parabola, depending on its orientation.
  • The equation of a parabola can be written in the form {(y-k)^2=4ap(x-h)$}$.
  • Parabolas have many real-world applications, including the design of satellite dishes, the shape of a thrown ball, and the trajectory of a projectile.

Final Thoughts

Understanding the graph of a parabola is crucial in mathematics, and it's essential to have a comprehensive knowledge of its characteristics. By addressing some of the most frequently asked questions related to the graph of a parabola, we have provided a solid foundation for further studies in mathematics. Whether you are a student or a professional, understanding the graph of a parabola is essential in mathematics and has many real-world applications.