Select The Function Equivalent To H ( X ) = X 2 + 8 X + 6 H(x) = X^2 + 8x + 6 H ( X ) = X 2 + 8 X + 6 . Show All Steps In Your Process.A) P ( X ) = ( X + 4 ) 2 P(x) = (x + 4)^2 P ( X ) = ( X + 4 ) 2 B) K ( X ) = ( X + 4 ) 2 + 10 K(x) = (x + 4)^2 + 10 K ( X ) = ( X + 4 ) 2 + 10 C) W ( X ) = ( X + 4 ) 2 − 10 W(x) = (x + 4)^2 - 10 W ( X ) = ( X + 4 ) 2 − 10

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In mathematics, functions are used to describe relationships between variables. When given a function, we may be asked to find an equivalent function that represents the same relationship. In this article, we will explore how to select the function equivalent to $h(x) = x^2 + 8x + 6$.

Understanding the Given Function

The given function is $h(x) = x^2 + 8x + 6$. This is a quadratic function, which means it can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Completing the Square

To find an equivalent function, we can use the method of completing the square. This involves rewriting the quadratic function in a form that allows us to easily identify the vertex of the parabola.

Step 1: Factor out the Coefficient of $x^2$

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

$$h(x) = (x^2 + 8x) + 6$$

Step 2: Add and Subtract the Square of Half the Coefficient of $x$

Next, we need to add and subtract the square of half the coefficient of $x$. The coefficient of $x$ is 8, so half of this is 4. The square of 4 is 16.

$$h(x) = (x^2 + 8x + 16) - 16 + 6$$

Step 3: Rewrite the Expression as a Perfect Square

Now, we can rewrite the expression as a perfect square:

$$h(x) = (x + 4)^2 - 10$$

Evaluating the Options

We are given three options for the function equivalent to $h(x) = x^2 + 8x + 6$. Let's evaluate each option to see which one matches our result.

Option A: $p(x) = (x + 4)^2$

This option is close, but it is missing the constant term. Our result had a constant term of -10, so this option is not correct.

Option B: $k(x) = (x + 4)^2 + 10$

This option is also close, but it has the wrong sign for the constant term. Our result had a constant term of -10, so this option is not correct.

Option C: $w(x) = (x + 4)^2 - 10$

This option matches our result exactly. Therefore, the function equivalent to $h(x) = x^2 + 8x + 6$ is:

$w(x) = (x + 4)^2 - 10$

Conclusion

In this article, we used the method of completing the square to find an equivalent function to $h(x) = x^2 + 8x + 6$. We evaluated three options and found that the function equivalent to $h(x)$ is $w(x) = (x + 4)^2 - 10$. This demonstrates the importance of understanding the method of completing the square and how it can be used to find equivalent functions.

Key Takeaways

• The method of completing the square can be used to find equivalent functions.
• To complete the square, factor out the coefficient of $x^2$, add and subtract the square of half the coefficient of $x$, and rewrite the expression as a perfect square.
• The function equivalent to $h(x) = x^2 + 8x + 6$ is $w(x) = (x + 4)^2 - 10$.

Further Reading

If you would like to learn more about completing the square and finding equivalent functions, I recommend checking out the following resources:

• Khan Academy: Completing the Square
• Mathway: Completing the Square
• Wolfram Alpha: Completing the Square

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In our previous article, we explored how to select the function equivalent to $h(x) = x^2 + 8x + 6$ using the method of completing the square. In this article, we will answer some frequently asked questions about this topic.

Q: What is the method of completing the square?

A: The method of completing the square is a technique used to rewrite a quadratic function in a form that allows us to easily identify the vertex of the parabola. It involves factoring out the coefficient of $x^2$, adding and subtracting the square of half the coefficient of $x$, and rewriting the expression as a perfect square.

Q: Why do we need to complete the square?

A: Completing the square allows us to easily identify the vertex of the parabola, which is the point where the function changes from increasing to decreasing or vice versa. This is useful in many applications, such as graphing functions and finding the maximum or minimum value of a function.

Q: How do I know which option is correct?

A: To determine which option is correct, you need to evaluate each option and see which one matches the result you obtained by completing the square. In our previous article, we evaluated three options and found that the function equivalent to $h(x) = x^2 + 8x + 6$ is $w(x) = (x + 4)^2 - 10$.

Q: Can I use completing the square to find the function equivalent to any quadratic function?

A: Yes, you can use completing the square to find the function equivalent to any quadratic function. However, you need to make sure that the quadratic function is in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square 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 evaluating all options to determine which one is correct

Q: How can I practice completing the square?

A: You can practice completing the square by working through examples and exercises. You can also use online resources, such as Khan Academy and Mathway, to practice completing the square.

Q: What are some real-world applications of completing the square?

A: Completing the square has many real-world applications, including:

• Graphing functions
• Finding the maximum or minimum value of a function
• Solving systems of equations
• Modeling real-world phenomena

Conclusion

In this article, we answered some frequently asked questions about selecting the function equivalent to $h(x) = x^2 + 8x + 6$ using the method of completing the square. We hope this article has been helpful in understanding this topic and has provided you with the information you need to practice completing the square.

Key Takeaways

• The method of completing the square is a technique used to rewrite a quadratic function in a form that allows us to easily identify the vertex of the parabola.
• Completing the square allows us to easily identify the vertex of the parabola, which is the point where the function changes from increasing to decreasing or vice versa.
• You can use completing the square to find the function equivalent to any quadratic function.
• Some common mistakes to avoid when completing the square include not factoring out the coefficient of $x^2$, not adding and subtracting the square of half the coefficient of $x$, and not rewriting the expression as a perfect square.

Further Reading

If you would like to learn more about completing the square and finding equivalent functions, we recommend checking out the following resources:

• Khan Academy: Completing the Square
• Mathway: Completing the Square
• Wolfram Alpha: Completing the Square

We hope this article has been helpful in understanding how to select the function equivalent to $h(x) = x^2 + 8x + 6$. If you have any questions or need further clarification, please don't hesitate to ask.