Which Of The Following Correctly Transforms $f(x)=x^2+9$ To $f(2x+1)$ For $ X = 2 X=2 X = 2 [/tex]?A. 26 B. 25 C. 14 D. 34

Introduction

In mathematics, transforming functions is a crucial concept that helps us understand how functions behave under different conditions. When we transform a function, we are essentially changing its input or output values to create a new function. In this article, we will explore how to transform the function $f(x)=x^2+9$ to $f(2x+1)$ and evaluate it for $x=2$. We will also discuss the importance of function transformation in mathematics and its applications.

Understanding Function Transformation

Function transformation is a process of changing the input or output values of a function to create a new function. This can be done in various ways, such as:

• Horizontal Shift: Shifting the function to the left or right by a certain value.
• Vertical Shift: Shifting the function up or down by a certain value.
• Horizontal Stretch: Stretching the function horizontally by a certain factor.
• Vertical Stretch: Stretching the function vertically by a certain factor.
• Reflection: Reflecting the function across the x-axis or y-axis.

Transforming $f(x)=x^2+9$ to $f(2x+1)$

To transform $f(x)=x^2+9$ to $f(2x+1)$, we need to replace $x$ with $2x+1$ in the original function. This means that we will substitute $2x+1$ for every instance of $x$ in the function.

$$f(2x+1)=(2x+1)^2+9$$

Expanding the Expression

To expand the expression, we need to use the binomial theorem, which states that:

$$(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$$

In this case, we have:

$$(2x+1)^2=(2x)^2+2(2x)(1)+1^2$$

$$=4x^2+4x+1$$

Substituting the Expression into the Function

Now that we have expanded the expression, we can substitute it into the function:

$$f(2x+1)=(4x^2+4x+1)+9$$

$$=4x^2+4x+10$$

Evaluating the Function for $x=2$

To evaluate the function for $x=2$, we need to substitute $2$ for $x$ in the function:

$$f(2(2)+1)=4(2)^2+4(2)+10$$

$$=16+8+10$$

$$=34$$

Conclusion

In conclusion, we have successfully transformed the function $f(x)=x^2+9$ to $f(2x+1)$ and evaluated it for $x=2$. The correct answer is $34$.

Importance of Function Transformation

Function transformation is an essential concept in mathematics that has numerous applications in various fields, such as:

• Calculus: Function transformation is used to find the derivative and integral of a function.
• Algebra: Function transformation is used to solve equations and inequalities.
• Geometry: Function transformation is used to describe the shape and size of geometric figures.
• Physics: Function transformation is used to describe the motion of objects and the behavior of physical systems.

Final Thoughts

Q: What is function transformation?

A: Function transformation is the process of changing the input or output values of a function to create a new function. This can be done in various ways, such as horizontal shift, vertical shift, horizontal stretch, vertical stretch, and reflection.

Q: Why is function transformation important?

A: Function transformation is an essential concept in mathematics that has numerous applications in various fields, such as calculus, algebra, geometry, and physics. It helps us understand how functions behave under different conditions and is used to solve a wide range of mathematical problems.

Q: How do I transform a function?

A: To transform a function, you need to replace the input variable with a new expression. For example, if you want to transform the function $f(x)=x^2+9$ to $f(2x+1)$, you need to substitute $2x+1$ for every instance of $x$ in the function.

Q: What are the different types of function transformations?

A: There are several types of function transformations, including:

• Horizontal Shift: Shifting the function to the left or right by a certain value.
• Vertical Shift: Shifting the function up or down by a certain value.
• Horizontal Stretch: Stretching the function horizontally by a certain factor.
• Vertical Stretch: Stretching the function vertically by a certain factor.
• Reflection: Reflecting the function across the x-axis or y-axis.

Q: How do I evaluate a transformed function?

A: To evaluate a transformed function, you need to substitute the given value of the input variable into the transformed function. For example, if you want to evaluate the function $f(2x+1)=4x^2+4x+10$ for $x=2$, you need to substitute $2$ for $x$ in the function.

Q: What are some common applications of function transformation?

A: Function transformation has numerous applications in various fields, including:

• Calculus: Function transformation is used to find the derivative and integral of a function.
• Algebra: Function transformation is used to solve equations and inequalities.
• Geometry: Function transformation is used to describe the shape and size of geometric figures.
• Physics: Function transformation is used to describe the motion of objects and the behavior of physical systems.

Q: How do I determine the correct answer when evaluating a transformed function?

A: To determine the correct answer when evaluating a transformed function, you need to follow the order of operations (PEMDAS) and substitute the given value of the input variable into the transformed function. You also need to simplify the expression and evaluate it to find the correct answer.

Q: What are some common mistakes to avoid when transforming functions?

A: Some common mistakes to avoid when transforming functions include:

• Not following the order of operations (PEMDAS)
• Not substituting the correct value of the input variable
• Not simplifying the expression
• Not evaluating the expression correctly

Q: How can I practice function transformation?

A: You can practice function transformation by working through examples and exercises in your textbook or online resources. You can also try transforming different types of functions, such as linear, quadratic, and polynomial functions.