Which Equation Represents The Vertex Form Of The Equation $y = X^2 - 6x + 11$?A. $y = (x - 6)^2 + 11$ B. $y = (x - 3)^2 + 2$ C. $y = (x - 3)^2 + 11$ D. $y = (x - 6)^2 + 2$

Introduction

In mathematics, a quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The standard form of a quadratic equation is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. However, there is another form of a quadratic equation known as the vertex form, which is represented as $y = a(x - h)^2 + k$. In this article, we will discuss the vertex form of a quadratic equation and provide a step-by-step guide on how to convert the standard form to the vertex form.

What is the Vertex Form of a Quadratic Equation?

The vertex form of a quadratic equation is a way to represent a quadratic function in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The vertex form is useful because it allows us to easily identify the vertex of the parabola, which is the maximum or minimum point of the function.

Converting Standard Form to Vertex Form

To convert the standard form of a quadratic equation to the vertex form, we need to complete the square. The process of completing the square involves rewriting the quadratic expression in a way that allows us to easily identify the vertex of the parabola.

Step 1: Write Down the Standard Form

The standard form of a quadratic equation is $y = ax^2 + bx + c$. In this case, we have $y = x^2 - 6x + 11$.

Step 2: Factor Out the Coefficient of $x^2$

The first step in completing the square is to factor out the coefficient of $x^2$, which is $a$. In this case, we have $a = 1$, so we can write:

$$y = 1(x^2 - 6x) + 11$$

Step 3: Add and Subtract the Square of Half the Coefficient of $x$

The next step is to add and subtract the square of half the coefficient of $x$. In this case, the coefficient of $x$ is $-6$, so we need to add and subtract $(-6/2)^2 = 9$:

$$y = 1(x^2 - 6x + 9 - 9) + 11$$

Step 4: Rewrite the Expression as a Perfect Square

Now we can rewrite the expression as a perfect square:

$$y = 1((x - 3)^2 - 9) + 11$$

Step 5: Simplify the Expression

Finally, we can simplify the expression by combining the constants:

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

Conclusion

In this article, we discussed the vertex form of a quadratic equation and provided a step-by-step guide on how to convert the standard form to the vertex form. We used the equation $y = x^2 - 6x + 11$ as an example and showed how to complete the square to obtain the vertex form $y = (x - 3)^2 + 2$. The vertex form is a useful way to represent a quadratic function, as it allows us to easily identify the vertex of the parabola.

Which Equation Represents the Vertex Form of the Equation $y = x^2 - 6x + 11$?

Based on the steps we followed to convert the standard form to the vertex form, we can see that the correct answer is:

• B. $y = (x - 3)^2 + 2$

This is because we obtained the vertex form $y = (x - 3)^2 + 2$ by completing the square.

Comparison of Options

Let's compare the options:

• A. $y = (x - 6)^2 + 11$: This is incorrect because we obtained the vertex form $y = (x - 3)^2 + 2$, not $y = (x - 6)^2 + 11$.
• B. $y = (x - 3)^2 + 2$: This is correct because we obtained the vertex form $y = (x - 3)^2 + 2$ by completing the square.
• C. $y = (x - 3)^2 + 11$: This is incorrect because we obtained the vertex form $y = (x - 3)^2 + 2$, not $y = (x - 3)^2 + 11$.
• D. $y = (x - 6)^2 + 2$: This is incorrect because we obtained the vertex form $y = (x - 3)^2 + 2$, not $y = (x - 6)^2 + 2$.

Conclusion

$$$$

Q&A: Vertex Form of a Quadratic Equation

Q: What is the vertex form of a quadratic equation?

A: The vertex form of a quadratic equation is a way to represent a quadratic function in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

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

A: To convert the standard form to the vertex form, you need to complete the square. The process of completing the square involves rewriting the quadratic expression in a way that allows you to easily identify the vertex of the parabola.

Q: What is completing the square?

A: Completing the square is a process of rewriting a quadratic expression in a way that allows you to easily identify the vertex of the parabola. It involves adding and subtracting a constant term to create a perfect square trinomial.

Q: How do I complete the square?

A: To complete the square, follow these steps:

1. Write down the standard form of the quadratic equation.
2. Factor out the coefficient of $x^2$.
3. Add and subtract the square of half the coefficient of $x$.
4. Rewrite the expression as a perfect square.
5. Simplify the expression.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the maximum or minimum point of the function. It is represented by the point $(h, k)$ in the vertex form of a quadratic 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 quadratic expression in the vertex form. The vertex is represented by the point $(h, k)$ in the vertex form.

Q: What are the advantages of using the vertex form?

A: The vertex form has several advantages, including:

• It allows you to easily identify the vertex of the parabola.
• It makes it easier to graph the parabola.
• It makes it easier to find the maximum or minimum point of the function.

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

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

• Not factoring out the coefficient of $x^2$.
• Not adding and subtracting the square of half the coefficient of $x$.
• Not rewriting the expression as a perfect square.
• Not simplifying the expression.

Q: How do I check my work when converting to the vertex form?

A: To check your work when converting to the vertex form, follow these steps:

1. Rewrite the quadratic expression in the standard form.
2. Compare the standard form with the vertex form.
3. Check that the vertex form is correct.

Conclusion

In conclusion, the vertex form of a quadratic equation is a useful way to represent a quadratic function. It allows you to easily identify the vertex of the parabola and makes it easier to graph the parabola. By following the steps outlined in this article, you can convert the standard form to the vertex form and check your work.

Frequently Asked Questions

• Q: What is the vertex form of a quadratic equation? A: The vertex form of a quadratic equation is a way to represent a quadratic function in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• Q: How do I convert the standard form to the vertex form? A: To convert the standard form to the vertex form, you need to complete the square.
• Q: What is completing the square? A: Completing the square is a process of rewriting a quadratic expression in a way that allows you to easily identify the vertex of the parabola.
• 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 quadratic expression in the vertex form.

Additional Resources

• Vertex Form of a Quadratic Equation: A comprehensive guide to the vertex form of a quadratic equation.
• Completing the Square: A step-by-step guide to completing the square.
• Vertex Form of a Quadratic Equation Worksheet: A worksheet with exercises to practice converting the standard form to the vertex form.