The Function H H H Is Defined As H ( X ) = 6 X 2 + 7 H(x) = 6x^2 + 7 H ( X ) = 6 X 2 + 7 . Find H ( X + 2 H(x+2 H ( X + 2 ]. Write Your Answer Without Parentheses, And Simplify It As Much As Possible. H ( X + 2 ) = 6 X 2 + 24 X + 31 H(x+2) = 6x^2 + 24x + 31 H ( X + 2 ) = 6 X 2 + 24 X + 31

Introduction

In mathematics, functions are used to describe the relationship between variables. The function $h(x) = 6x^2 + 7$ is a quadratic function that represents a parabola. In this article, we will explore the transformation of this function by finding $h(x+2)$.

Understanding the Function h(x)

The function $h(x) = 6x^2 + 7$ is a quadratic function that can be written in the form $h(x) = ax^2 + bx + c$, where $a = 6$, $b = 0$, and $c = 7$. This function represents a parabola that opens upwards, with its vertex at the point $(0, 7)$.

Finding h(x+2)

To find $h(x+2)$, we need to substitute $(x+2)$ into the function $h(x) = 6x^2 + 7$. This means that we will replace every instance of $x$ with $(x+2)$.

Step 1: Substitute (x+2) into the Function

$h(x+2) = 6(x+2)^2 + 7$

Step 2: Expand the Expression

To expand the expression, we need to use the formula $(a+b)^2 = a^2 + 2ab + b^2$. In this case, $a = x$ and $b = 2$.

$h(x+2) = 6(x^2 + 4x + 4) + 7$

Step 3: Simplify the Expression

Now, we can simplify the expression by distributing the $6$ to each term inside the parentheses.

$h(x+2) = 6x^2 + 24x + 24 + 7$

Step 4: Combine Like Terms

Finally, we can combine like terms by adding the constants $24$ and $7$.

$h(x+2) = 6x^2 + 24x + 31$

Conclusion

In this article, we have found the transformation of the function $h(x) = 6x^2 + 7$ by finding $h(x+2)$. We have used the substitution method to replace every instance of $x$ with $(x+2)$ and then expanded and simplified the expression. The final result is $h(x+2) = 6x^2 + 24x + 31$.

Key Takeaways

• The function $h(x) = 6x^2 + 7$ is a quadratic function that represents a parabola.
• To find $h(x+2)$, we need to substitute $(x+2)$ into the function $h(x) = 6x^2 + 7$.
• We can use the substitution method to replace every instance of $x$ with $(x+2)$ and then expand and simplify the expression.
• The final result is $h(x+2) = 6x^2 + 24x + 31$.

Further Reading

If you want to learn more about quadratic functions and their transformations, we recommend checking out the following resources:

• Khan Academy: Quadratic Functions
• Math Is Fun: Quadratic Functions
• Wolfram MathWorld: Quadratic Function

References

• [1] Khan Academy. (n.d.). Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/quadratic-equations
• [2] Math Is Fun. (n.d.). Quadratic Functions. Retrieved from https://www.mathisfun.com/algebra/quadratic-functions.html
• [3] Wolfram MathWorld. (n.d.). Quadratic Function. Retrieved from https://mathworld.wolfram.com/QuadraticFunction.html
  The Function h(x) and Its Transformations: Q&A =====================================================

Introduction

In our previous article, we explored the transformation of the function $h(x) = 6x^2 + 7$ by finding $h(x+2)$. In this article, we will answer some frequently asked questions about the function $h(x)$ and its transformations.

Q&A

Q: What is the function h(x) = 6x^2 + 7?

A: The function $h(x) = 6x^2 + 7$ is a quadratic function that represents a parabola. It is a polynomial function of degree 2, where $a = 6$, $b = 0$, and $c = 7$.

Q: What is the vertex of the parabola represented by the function h(x) = 6x^2 + 7?

A: The vertex of the parabola represented by the function $h(x) = 6x^2 + 7$ is the point $(0, 7)$.

Q: How do I find h(x+2)?

A: To find $h(x+2)$, you need to substitute $(x+2)$ into the function $h(x) = 6x^2 + 7$. This means that you will replace every instance of $x$ with $(x+2)$.

Q: What is the final result of finding h(x+2)?

A: The final result of finding $h(x+2)$ is $h(x+2) = 6x^2 + 24x + 31$.

Q: What is the difference between h(x) and h(x+2)?

A: The difference between $h(x)$ and $h(x+2)$ is that $h(x+2)$ is a transformation of the function $h(x)$, where every instance of $x$ is replaced with $(x+2)$.

Q: Can I use the same method to find h(x+3) or h(x+4)?

A: Yes, you can use the same method to find $h(x+3)$ or $h(x+4)$. Simply substitute $(x+3)$ or $(x+4)$ into the function $h(x) = 6x^2 + 7$ and follow the same steps as before.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have many real-world applications, such as:

• Modeling the trajectory of a projectile
• Describing the motion of an object under the influence of gravity
• Finding the maximum or minimum value of a function
• Solving optimization problems

Q: How can I learn more about quadratic functions and their transformations?

A: You can learn more about quadratic functions and their transformations by:

• Checking out online resources such as Khan Academy, Math Is Fun, and Wolfram MathWorld
• Reading books on algebra and geometry
• Practicing problems and exercises to reinforce your understanding
• Seeking help from a teacher or tutor

Conclusion

In this article, we have answered some frequently asked questions about the function $h(x)$ and its transformations. We hope that this article has been helpful in clarifying any doubts you may have had about quadratic functions and their transformations.

Key Takeaways

• The function $h(x) = 6x^2 + 7$ is a quadratic function that represents a parabola.
• To find $h(x+2)$, you need to substitute $(x+2)$ into the function $h(x) = 6x^2 + 7$.
• The final result of finding $h(x+2)$ is $h(x+2) = 6x^2 + 24x + 31$.
• Quadratic functions have many real-world applications, such as modeling the trajectory of a projectile and describing the motion of an object under the influence of gravity.

Further Reading

If you want to learn more about quadratic functions and their transformations, we recommend checking out the following resources:

• Khan Academy: Quadratic Functions
• Math Is Fun: Quadratic Functions
• Wolfram MathWorld: Quadratic Function

References

• [1] Khan Academy. (n.d.). Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/quadratic-equations
• [2] Math Is Fun. (n.d.). Quadratic Functions. Retrieved from https://www.mathisfun.com/algebra/quadratic-functions.html
• [3] Wolfram MathWorld. (n.d.). Quadratic Function. Retrieved from https://mathworld.wolfram.com/QuadraticFunction.html