The Vertex Form Of The Equation Of A Parabola Is Y = ( X − 3 ) 2 + 36 Y = (x-3)^2 + 36 Y = ( X − 3 ) 2 + 36 . What Is The Standard Form Of The Equation?A. Y = X 2 − 6 X + 45 Y = X^2 - 6x + 45 Y = X 2 − 6 X + 45 B. Y = 3 X 2 − 6 X + 45 Y = 3x^2 - 6x + 45 Y = 3 X 2 − 6 X + 45 C. Y = X 2 + 6 X + 36 Y = X^2 + 6x + 36 Y = X 2 + 6 X + 36 D. Y = X 2 + X + 18 Y = X^2 + X + 18 Y = X 2 + X + 18

Introduction

The vertex form of a parabola is a powerful tool used to describe the shape and position of a parabola on a coordinate plane. It is represented by the equation $y = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola. In this article, we will explore the standard form of the equation of a parabola and how it can be obtained from the vertex form.

Understanding the Vertex Form

The vertex form of a parabola is given by the equation $y = a(x-h)^2 + k$. Here, $a$ is the coefficient of the squared term, $h$ is the x-coordinate of the vertex, and $k$ is the y-coordinate of the vertex. The vertex form is useful because it allows us to easily identify the vertex of the parabola and the direction it opens.

Converting to Standard Form

To convert the vertex form of a parabola to the standard form, we need to expand the squared term and simplify the equation. The standard form of a parabola is given by the equation $y = ax^2 + bx + c$. To obtain the standard form from the vertex form, we need to expand the squared term and combine like terms.

Example: Converting the Vertex Form to Standard Form

Let's consider the vertex form of a parabola: $y = (x-3)^2 + 36$. To convert this to the standard form, we need to expand the squared term and simplify the equation.

Step 1: Expand the Squared Term

To expand the squared term, we need to use the formula $(x-h)^2 = x^2 - 2hx + h^2$. In this case, $h = 3$, so we have:

$$(x-3)^2 = x^2 - 2(3)x + 3^2$$

Step 2: Simplify the Equation

Now, we can simplify the equation by combining like terms:

$$y = x^2 - 6x + 9 + 36$$

$$y = x^2 - 6x + 45$$

Therefore, the standard form of the equation is $y = x^2 - 6x + 45$.

Conclusion

In this article, we have explored the vertex form of a parabola and how it can be converted to the standard form. We have seen that the standard form of a parabola is given by the equation $y = ax^2 + bx + c$, and that it can be obtained from the vertex form by expanding the squared term and simplifying the equation. We have also seen an example of how to convert the vertex form to the standard form.

Answer

The standard form of the equation of a parabola is $y = x^2 - 6x + 45$.

Discussion

This problem is a great example of how to convert the vertex form of a parabola to the standard form. It requires the use of algebraic techniques, such as expanding squared terms and combining like terms. The standard form of a parabola is useful because it allows us to easily identify the vertex of the parabola and the direction it opens.

Related Topics

• Vertex Form of a Parabola: The vertex form of a parabola is a powerful tool used to describe the shape and position of a parabola on a coordinate plane.
• Standard Form of a Parabola: The standard form of a parabola is given by the equation $y = ax^2 + bx + c$.
• Expanding Squared Terms: Expanding squared terms is an important algebraic technique used to convert the vertex form of a parabola to the standard form.

References

• Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships.
• Geometry: Geometry is a branch of mathematics that deals with the study of shapes and their properties.
• Calculus: Calculus is a branch of mathematics that deals with the study of rates of change and accumulation.

Further Reading

• Vertex Form of a Parabola: The vertex form of a parabola is a powerful tool used to describe the shape and position of a parabola on a coordinate plane.
• Standard Form of a Parabola: The standard form of a parabola is given by the equation $y = ax^2 + bx + c$.
• Expanding Squared Terms: Expanding squared terms is an important algebraic technique used to convert the vertex form of a parabola to the standard form.
  Q&A: The Vertex Form of a Parabola =====================================

Q: What is the vertex form of a parabola?

A: The vertex form of a parabola is a powerful tool used to describe the shape and position of a parabola on a coordinate plane. It is represented by the equation $y = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola.

Q: What is the standard form of a parabola?

A: The standard form of a parabola is given by the equation $y = ax^2 + bx + c$. It is obtained from the vertex form by expanding the squared term and simplifying the equation.

Q: How do I convert the vertex form to the standard form?

A: To convert the vertex form to the standard form, you need to expand the squared term and simplify the equation. This involves using algebraic techniques such as expanding squared terms and combining like terms.

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

A: The vertex form of a parabola is significant because it allows us to easily identify the vertex of the parabola and the direction it opens. This is useful in a variety of applications, including physics, engineering, and computer science.

Q: Can you give an example of how to convert the vertex form to the standard form?

A: Let's consider the vertex form of a parabola: $y = (x-3)^2 + 36$. To convert this to the standard form, we need to expand the squared term and simplify the equation.

Step 1: Expand the Squared Term

To expand the squared term, we need to use the formula $(x-h)^2 = x^2 - 2hx + h^2$. In this case, $h = 3$, so we have:

$$(x-3)^2 = x^2 - 2(3)x + 3^2$$

Step 2: Simplify the Equation

Now, we can simplify the equation by combining like terms:

$$y = x^2 - 6x + 9 + 36$$

$$y = x^2 - 6x + 45$$

Therefore, the standard form of the equation is $y = x^2 - 6x + 45$.

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

A: Some common mistakes to avoid when converting the vertex form to the standard form include:

• Not expanding the squared term correctly: Make sure to use the formula $(x-h)^2 = x^2 - 2hx + h^2$ to expand the squared term.
• Not combining like terms correctly: Make sure to combine like terms carefully to simplify the equation.
• Not checking the equation for errors: Make sure to check the equation for errors before considering it to be correct.

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

A: The vertex form of a parabola has a variety of real-world applications, including:

• Physics: The vertex form of a parabola is used to describe the motion of objects under the influence of gravity.
• Engineering: The vertex form of a parabola is used to design curves and surfaces for a variety of applications, including architecture and product design.
• Computer Science: The vertex form of a parabola is used in computer graphics to create smooth curves and surfaces.

Q: Can you recommend any resources for learning more about the vertex form of a parabola?

A: Yes, there are a variety of resources available for learning more about the vertex form of a parabola, including:

• Textbooks: There are many textbooks available that cover the topic of the vertex form of a parabola, including "Algebra" by Michael Artin and "Calculus" by Michael Spivak.
• Online Resources: There are many online resources available that cover the topic of the vertex form of a parabola, including Khan Academy and MIT OpenCourseWare.
• Video Lectures: There are many video lectures available that cover the topic of the vertex form of a parabola, including those on YouTube and Coursera.