Select The Correct Answer.The Vertex Of A Parabola Is At The Point (3, 1), And Its Focus Is At (3, 5). What Function Does The Graph Represent?A. F ( X ) = 1 1 ( X − 3 ) 2 − 1 F(x)=\frac{1}{1}(x-3)^2-1 F ( X ) = 1 1 ​ ( X − 3 ) 2 − 1 B. F ( X ) = 1 4 ( X + 3 ) 2 − 1 F(x)=\frac{1}{4}(x+3)^2-1 F ( X ) = 4 1 ​ ( X + 3 ) 2 − 1 C.

Introduction

Parabolas are a fundamental concept in mathematics, and understanding their properties is crucial for various applications in science, engineering, and other fields. In this article, we will delve into the world of parabolas, focusing on identifying the correct function that represents a given parabola. We will explore the properties of parabolas, including the vertex and focus, and use this knowledge to select the correct answer from a set of given functions.

Properties of Parabolas

A parabola is a quadratic function that can be represented in the form $f(x) = ax^2 + bx + c$. The vertex of a parabola is the point where the parabola changes direction, and it is represented by the coordinates $(h, k)$. The focus of a parabola is a fixed point that is equidistant from the vertex and the directrix. The directrix is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of $p$ from the vertex.

Vertex and Focus

In this problem, we are given that the vertex of the parabola is at the point $(3, 1)$ and the focus is at $(3, 5)$. This information allows us to determine the value of $p$, which is the distance between the vertex and the focus. Since the focus is at $(3, 5)$ and the vertex is at $(3, 1)$, the value of $p$ is $5 - 1 = 4$.

Equation of a Parabola

The equation of a parabola can be represented in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. Since we know that the vertex is at $(3, 1)$, we can substitute these values into the equation to get $f(x) = a(x - 3)^2 + 1$.

Determining the Value of $a$

To determine the value of $a$, we need to use the information about the focus. Since the focus is at $(3, 5)$, we know that the distance between the vertex and the focus is $4$. This means that the value of $p$ is $4$, and we can use this information to determine the value of $a$.

The equation of a parabola can be represented in the form $f(x) = a(x - h)^2 + k + p$. Since we know that the vertex is at $(3, 1)$ and the focus is at $(3, 5)$, we can substitute these values into the equation to get $f(x) = a(x - 3)^2 + 1 + 4$. Simplifying this equation, we get $f(x) = a(x - 3)^2 + 5$.

Comparing the Options

Now that we have the equation of the parabola, we can compare it to the given options to determine the correct function. The options are:

A. $f(x)=\frac{1}{1}(x-3)^2-1$ B. $f(x)=\frac{1}{4}(x+3)^2-1$ C. $f(x)=\frac{1}{4}(x-3)^2+5$

Comparing the equation we derived to the options, we can see that option C is the correct function. The equation $f(x) = \frac{1}{4}(x - 3)^2 + 5$ matches the equation we derived, and it represents the parabola with the given vertex and focus.

Conclusion

In this article, we explored the properties of parabolas, including the vertex and focus. We used this knowledge to determine the correct function that represents a given parabola. By comparing the equation we derived to the given options, we were able to select the correct function. This problem demonstrates the importance of understanding the properties of parabolas and how to apply this knowledge to solve problems.

Final Answer

The final answer is:

Q&A: Understanding Parabolas

In our previous article, we explored the properties of parabolas, including the vertex and focus. We also used this knowledge to determine the correct function that represents a given parabola. In this article, we will continue to delve into the world of parabolas, answering some of the most frequently asked questions about these fascinating curves.

Q: What is a parabola?

A parabola is a quadratic function that can be represented in the form $f(x) = ax^2 + bx + c$. It is a U-shaped curve that opens upwards or downwards, and it has a single turning point, known as the vertex.

Q: What is the vertex of a parabola?

The vertex of a parabola is the point where the parabola changes direction. It is represented by the coordinates $(h, k)$, where $h$ is the x-coordinate and $k$ is the y-coordinate.

Q: What is the focus of a parabola?

The focus of a parabola is a fixed point that is equidistant from the vertex and the directrix. The directrix is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of $p$ from the vertex.

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

To determine the value of $a$ in a parabola, you need to use the information about the focus. Since the focus is at $(3, 5)$, we know that the distance between the vertex and the focus is $4$. This means that the value of $p$ is $4$, and we can use this information to determine the value of $a$.

Q: What is the equation of a parabola?

The equation of a parabola can be represented in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: How do I compare the options to determine the correct function?

To compare the options to determine the correct function, you need to substitute the values of the vertex and focus into the equation of the parabola. This will give you the correct function that represents the parabola.

Q: What is the final answer?

The final answer is:

C. $f(x)=\frac{1}{4}(x-3)^2+5$

Common Mistakes to Avoid

When working with parabolas, there are several common mistakes to avoid. These include:

• Not using the correct equation of the parabola
• Not substituting the values of the vertex and focus into the equation
• Not comparing the options to determine the correct function
• Not using the correct value of $a$

Conclusion

In this article, we have answered some of the most frequently asked questions about parabolas. We have explored the properties of parabolas, including the vertex and focus, and we have used this knowledge to determine the correct function that represents a given parabola. By avoiding common mistakes and using the correct equation of the parabola, you can ensure that you get the correct answer.

Additional Resources

If you are looking for additional resources to help you understand parabolas, there are several online resources available. These include:

• Khan Academy: Parabolas
• Mathway: Parabolas
• Wolfram Alpha: Parabolas

Final Tips

When working with parabolas, it is essential to use the correct equation and to substitute the values of the vertex and focus into the equation. By doing so, you can ensure that you get the correct answer. Additionally, make sure to compare the options to determine the correct function, and avoid common mistakes such as not using the correct value of $a$.