The Standard Form Of The Equation Of A Parabola Is X = Y 2 + 6 Y + 1 X = Y^2 + 6y + 1 X = Y 2 + 6 Y + 1 . What Is The Vertex Form Of The Equation?A. X = ( Y + 3 ) 2 − 5 X = (y + 3)^2 - 5 X = ( Y + 3 ) 2 − 5 B. X = ( Y + 6 ) 2 − 35 X = (y + 6)^2 - 35 X = ( Y + 6 ) 2 − 35 C. X = ( Y + 6 ) 2 − 11 X = (y + 6)^2 - 11 X = ( Y + 6 ) 2 − 11 D. X = ( Y + 3 ) 2 − 8 X = (y + 3)^2 - 8 X = ( Y + 3 ) 2 − 8

Mar 13, 2025 by ADMIN 396 views

**The Standard Form of a Parabola: Converting to Vertex Form**

Introduction

In mathematics, a parabola is a quadratic equation that can be represented in various forms. The standard form of a parabola is given by the equation $x = y^2 + 6y + 1$ . However, this form can be converted to the vertex form, which provides valuable information about the parabola's vertex and axis of symmetry. In this article, we will explore the process of converting the standard form of a parabola to its vertex form.

Understanding the Standard Form

The standard form of a parabola is given by the equation $x = y^2 + 6y + 1$ . This equation represents a quadratic function in the form of $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. In this case, the equation is in terms of $y$ , and the variable $x$ is the dependent variable.

Completing the Square

To convert the standard form of a parabola to its vertex form, we need to complete the square. This involves rewriting the equation in a form that allows us to easily identify the vertex and axis of symmetry. To complete the square, we start by factoring out the coefficient of the $y^2$ term, which is 1 in this case.

x = (y^2 + 6y) + 1

Next, we add and subtract the square of half the coefficient of the $y$ term inside the parentheses. In this case, the coefficient of the $y$ term is 6, so we add and subtract $(6/2)^2 = 9$ inside the parentheses.

x = (y^2 + 6y + 9) - 9 + 1

Now, we can rewrite the equation as a perfect square trinomial.

x = (y + 3)^2 - 8

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. In this case, we have $a = 1$ , $k = -3$ , and $h = -8$ . Therefore, the vertex form of the equation is $x = (y + 3)^2 - 8$ .

Conclusion

In conclusion, we have successfully converted the standard form of a parabola to its vertex form. The vertex form provides valuable information about the parabola's vertex and axis of symmetry, which can be useful in various applications. By completing the square, we were able to rewrite the equation in a form that allows us to easily identify the vertex and axis of symmetry.

Answer

The correct answer is D. $x = (y + 3)^2 - 8$ .

Discussion

This problem requires the student to understand the concept of completing the square and converting the standard form of a parabola to its vertex form. The student should be able to identify the vertex and axis of symmetry of the parabola and write the equation in vertex form.

Practice Problems

Convert the standard form of a parabola to its vertex form: $x = y^2 + 4y + 4$
Find the vertex and axis of symmetry of the parabola: $x = (y - 2)^2 + 3$
Convert the vertex form of a parabola to its standard form: $x = (y + 1)^2 - 2$

References

[1] Algebra and Trigonometry, Michael Sullivan, 10th edition
[2] Calculus, James Stewart, 7th edition
[3] Mathematics for the Nonmathematician, Morris Kline, 1st edition
The Standard Form of a Parabola: Q&A =====================================

Q: What is the standard form of a parabola?

A: The standard form of a parabola is given by the equation $x = y^2 + 6y + 1$ . This equation represents a quadratic function in the form of $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Q: What is the vertex form of a parabola?

A: 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.

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

A: To convert the standard form of a parabola to its vertex form, you need to complete the square. This involves rewriting the equation in a form that allows you to easily identify the vertex and axis of symmetry.

Q: What is completing the square?

A: Completing the square is a process of rewriting a quadratic equation in a form that allows you to easily identify the vertex and axis of symmetry. It involves adding and subtracting a constant term inside the parentheses to create a perfect square trinomial.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

Factor out the coefficient of the $y^2$ term.
Add and subtract the square of half the coefficient of the $y$ term inside the parentheses.
Rewrite the equation as a perfect square trinomial.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is represented by the point $(h, k)$ in the vertex form of the equation.

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

A: To find the vertex of a parabola, you need to complete the square and rewrite the equation in vertex form. The vertex is represented by the point $(h, k)$ in the vertex form of the equation.

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

A: The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. It is represented by the equation $x = h$ in the vertex form of the equation.

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 complete the square and rewrite the equation in vertex form. The axis of symmetry is represented by the equation $x = h$ in the vertex form of the equation.

Q: What are some common mistakes to avoid when converting the standard form of a parabola to its vertex form?

A: Some common mistakes to avoid when converting the standard form of a parabola to its vertex form include:

Not factoring out the coefficient of the $y^2$ term.
Not adding and subtracting the square of half the coefficient of the $y$ term inside the parentheses.
Not rewriting the equation as a perfect square trinomial.
Not identifying the vertex and axis of symmetry correctly.

Q: How can I practice converting the standard form of a parabola to its vertex form?

A: You can practice converting the standard form of a parabola to its vertex form by working through examples and exercises. You can also use online resources and practice problems to help you improve your skills.

Q: What are some real-world applications of the standard form and vertex form of a parabola?

A: The standard form and vertex form of a parabola have many real-world applications, including:

Modeling the trajectory of a projectile.
Describing the shape of a parabolic mirror.
Analyzing the motion of an object under the influence of gravity.
Designing optical systems.

Q: How can I use the standard form and vertex form of a parabola in my career?

A: The standard form and vertex form of a parabola can be used in a variety of careers, including:

Physics and engineering.
Computer science and programming.
Mathematics and statistics.
Data analysis and visualization.

Q: What are some common misconceptions about the standard form and vertex form of a parabola?

A: Some common misconceptions about the standard form and vertex form of a parabola include:

Thinking that the standard form and vertex form are interchangeable.
Believing that the standard form is always easier to work with than the vertex form.
Assuming that the vertex form is only used in advanced mathematics and physics.

Q: How can I overcome these misconceptions and improve my understanding of the standard form and vertex form of a parabola?

A: You can overcome these misconceptions and improve your understanding of the standard form and vertex form of a parabola by:

Practicing and working through examples and exercises.
Reading and studying the subject matter.
Seeking help and guidance from teachers and mentors.
Exploring real-world applications and examples.