The Vertex Form Of The Equation Of A Parabola Is Y = 6 ( X − 2 ) 2 − 8 Y=6(x-2)^2-8 Y = 6 ( X − 2 ) 2 − 8 . What Is The Standard Form Of The Equation?A. Y = 6 X 2 − 4 X + 4 Y=6x^2-4x+4 Y = 6 X 2 − 4 X + 4 B. Y = 12 X 2 − 6 X + 8 Y=12x^2-6x+8 Y = 12 X 2 − 6 X + 8 C. Y = 6 X 2 − 24 X + 16 Y=6x^2-24x+16 Y = 6 X 2 − 24 X + 16 D. Y = 12 X 2 − 12 X + 16 Y=12x^2-12x+16 Y = 12 X 2 − 12 X + 16

Introduction

The vertex form of a parabola is a powerful tool in algebra, allowing us to easily identify the vertex of a parabola and its direction. However, in many cases, we need to convert the vertex form into the standard form, which is a more traditional and widely used form of a parabola's equation. In this article, we will explore how to convert the vertex form of a parabola into its standard form, using the given equation $y=6(x-2)^2-8$ as an example.

Understanding the Vertex Form

The vertex form of a parabola is given by the equation $y=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola. In the given equation $y=6(x-2)^2-8$, we can see that the vertex is at the point $(2,-8)$. The coefficient $a$ represents the direction and the width of the parabola.

Converting to Standard Form

To convert the vertex form into the standard form, we need to expand the squared term and simplify the equation. Let's start by expanding the squared term:

$$y=6(x-2)^2-8$$

$$y=6(x^2-4x+4)-8$$

$$y=6x^2-24x+24-8$$

$$y=6x^2-24x+16$$

The Standard Form

The standard form of a parabola is given by the equation $y=ax^2+bx+c$, where $a$, $b$, and $c$ are constants. In the given equation $y=6x^2-24x+16$, we can see that the standard form is:

$$y=6x^2-24x+16$$

Comparing with the Options

Now that we have the standard form of the equation, let's compare it with the given options:

A. $y=6x^2-4x+4$ B. $y=12x^2-6x+8$ C. $y=6x^2-24x+16$ D. $y=12x^2-12x+16$

We can see that option C matches our standard form, which is:

$$y=6x^2-24x+16$$

Conclusion

In this article, we have explored how to convert the vertex form of a parabola into its standard form. We used the given equation $y=6(x-2)^2-8$ as an example and expanded the squared term to simplify the equation. We then compared the resulting standard form with the given options and found that option C matches our result. This demonstrates the importance of understanding the vertex form and how to convert it into the standard form, which is a more traditional and widely used form of a parabola's equation.

Key Takeaways

• The vertex form of a parabola is given by the equation $y=a(x-h)^2+k$.
• To convert the vertex form into 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 $y=ax^2+bx+c$.
• Understanding the vertex form and how to convert it into the standard form is essential in algebra and mathematics.

Further Reading

For further reading on the vertex form and standard form of a parabola, we recommend the following resources:

• Khan Academy: Vertex Form of a Parabola
• Mathway: Vertex Form to Standard Form
• Wolfram MathWorld: Parabola

References

• [1] Khan Academy. (n.d.). Vertex Form of a Parabola. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f4c7/x2f6f4c8a/x2f6f4c8b
• [2] Mathway. (n.d.). Vertex Form to Standard Form. Retrieved from https://www.mathway.com/subjects/vertex-form-to-standard-form
• [3] Wolfram MathWorld. (n.d.). Parabola. Retrieved from https://mathworld.wolfram.com/Parabola.html
  The Vertex Form of a Parabola: Q&A =====================================

Introduction

In our previous article, we explored the vertex form of a parabola and how to convert it into the standard form. In this article, we will answer some frequently asked questions about the vertex form and standard form of a parabola.

Q: What is the vertex form of a parabola?

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

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

A: To find the vertex of a parabola in vertex form, you need to identify the values of $h$ and $k$ in the equation $y=a(x-h)^2+k$. The vertex is given by the point $(h,k)$.

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

A: To convert the vertex form into the standard form, you need to expand the squared term and simplify the equation. This involves multiplying out the squared term and combining like terms.

Q: What is the standard form of a parabola?

A: The standard form of a parabola is given by the equation $y=ax^2+bx+c$, where $a$, $b$, and $c$ are constants.

Q: How do I identify the direction and width of a parabola in vertex form?

A: To identify the direction and width of a parabola in vertex form, you need to look at the value of $a$ in the equation $y=a(x-h)^2+k$. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards. The width of the parabola is given by the value of $2a$.

Q: Can I convert a parabola from standard form to vertex form?

A: Yes, you can convert a parabola from standard form to vertex form by completing the square. This involves rewriting the equation in the form $y=a(x-h)^2+k$.

Q: What are some common mistakes to avoid when working with the vertex form and standard form of a parabola?

A: Some common mistakes to avoid when working with the vertex form and standard form of a parabola include:

• Not identifying the vertex correctly
• Not expanding the squared term correctly
• Not combining like terms correctly
• Not identifying the direction and width of the parabola correctly

Q: How do I use the vertex form and standard form of a parabola in real-world applications?

A: The vertex form and standard form of a parabola are used in a variety of real-world applications, including:

• Modeling the trajectory of a projectile
• Describing the shape of a parabolic mirror
• Finding the maximum or minimum value of a function

Conclusion

In this article, we have answered some frequently asked questions about the vertex form and standard form of a parabola. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of these important concepts.

Key Takeaways

• The vertex form of a parabola is given by the equation $y=a(x-h)^2+k$.
• To find the vertex of a parabola in vertex form, you need to identify the values of $h$ and $k$.
• To convert the vertex form into the standard form, you need to expand the squared term and simplify the equation.
• The standard form of a parabola is given by the equation $y=ax^2+bx+c$.
• Understanding the vertex form and standard form of a parabola is essential in algebra and mathematics.

Further Reading

For further reading on the vertex form and standard form of a parabola, we recommend the following resources:

• Khan Academy: Vertex Form of a Parabola
• Mathway: Vertex Form to Standard Form
• Wolfram MathWorld: Parabola

References

• [1] Khan Academy. (n.d.). Vertex Form of a Parabola. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f4c7/x2f6f4c8a/x2f6f4c8b
• [2] Mathway. (n.d.). Vertex Form to Standard Form. Retrieved from https://www.mathway.com/subjects/vertex-form-to-standard-form
• [3] Wolfram MathWorld. (n.d.). Parabola. Retrieved from https://mathworld.wolfram.com/Parabola.html