Which Function Results After Applying The Sequence Of Transformations, In This Order, To $f(x) = X^5$?1. Stretch Vertically By 32. Translate Up 1 Unit3. Translate Left 2 UnitsA. $f(x) = 3(x-2)^5 + 1$ B. $f(x) = 3(x+2)^5 + 1$

Introduction

In mathematics, functions can be transformed in various ways to create new functions. These transformations can be applied to functions in different orders, resulting in different final functions. In this article, we will explore the sequence of transformations applied to the function $f(x) = x^5$ and determine the resulting function.

The Original Function

The original function is given as $f(x) = x^5$. This is a basic polynomial function with a degree of 5.

Step 1: Stretch Vertically by 32

The first transformation applied to the function is a vertical stretch by a factor of 32. This means that the function will be multiplied by 32.

$f(x) = 32x^5$

Step 2: Translate Up 1 Unit

The second transformation applied to the function is a translation up 1 unit. This means that the function will be shifted upwards by 1 unit.

$f(x) = 32x^5 + 1$

Step 3: Translate Left 2 Units

The third and final transformation applied to the function is a translation left 2 units. This means that the function will be shifted to the left by 2 units.

$f(x) = 32(x-2)^5 + 1$

The Final Function

After applying the sequence of transformations in the given order, the resulting function is:

$f(x) = 32(x-2)^5 + 1$

This is the final function after applying the transformations.

Comparison with Options

Let's compare the resulting function with the given options:

A. $f(x) = 3(x-2)^5 + 1$ B. $f(x) = 3(x+2)^5 + 1$

The resulting function $f(x) = 32(x-2)^5 + 1$ is different from both options A and B. Option A has a different coefficient (3 instead of 32) and option B has a different translation (left 2 units instead of right 2 units).

Conclusion

In conclusion, after applying the sequence of transformations in the given order to the function $f(x) = x^5$, the resulting function is $f(x) = 32(x-2)^5 + 1$. This function is a result of a vertical stretch by 32, a translation up 1 unit, and a translation left 2 units.

Key Takeaways

• The sequence of transformations applied to a function can result in a different final function.
• The order of transformations is important and can affect the final function.
• A vertical stretch by a factor of 32, a translation up 1 unit, and a translation left 2 units result in the function $f(x) = 32(x-2)^5 + 1$.

Further Reading

For more information on function transformations, please refer to the following resources:

• Khan Academy: Function Transformations
• Math Is Fun: Function Transformations
• Wolfram MathWorld: Function Transformations
  Function Transformations: A Q&A Guide =====================================

Introduction

In our previous article, we explored the sequence of transformations applied to the function $f(x) = x^5$ and determined the resulting function. In this article, we will answer some frequently asked questions about function transformations.

Q: What is a function transformation?

A function transformation is a change made to a function to create a new function. These transformations can be applied to functions in different orders, resulting in different final functions.

Q: What are the different types of function transformations?

There are several types of function transformations, including:

• Vertical stretch: A vertical stretch by a factor of $a$ is a transformation that multiplies the function by $a$.
• Vertical compression: A vertical compression by a factor of $a$ is a transformation that multiplies the function by $\frac{1}{a}$.
• Horizontal stretch: A horizontal stretch by a factor of $a$ is a transformation that multiplies the input of the function by $a$.
• Horizontal compression: A horizontal compression by a factor of $a$ is a transformation that multiplies the input of the function by $\frac{1}{a}$.
• Translation: A translation is a transformation that shifts the function to the left or right by a certain number of units.

Q: How do I apply a vertical stretch to a function?

To apply a vertical stretch to a function, you multiply the function by the desired factor. For example, if you want to apply a vertical stretch by a factor of 3 to the function $f(x) = x^2$, you would get:

$f(x) = 3x^2$

Q: How do I apply a horizontal stretch to a function?

To apply a horizontal stretch to a function, you multiply the input of the function by the desired factor. For example, if you want to apply a horizontal stretch by a factor of 2 to the function $f(x) = x^2$, you would get:

$f(x) = (2x)^2$

Q: How do I apply a translation to a function?

To apply a translation to a function, you add or subtract the desired number of units from the input of the function. For example, if you want to apply a translation left by 2 units to the function $f(x) = x^2$, you would get:

$f(x) = (x-2)^2$

Q: What is the order of operations for function transformations?

The order of operations for function transformations is as follows:

1. Parentheses: Evaluate any expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I determine the resulting function after applying a sequence of transformations?

To determine the resulting function after applying a sequence of transformations, you apply each transformation in the given order. For example, if you want to apply a vertical stretch by a factor of 3, a translation up 1 unit, and a translation left 2 units to the function $f(x) = x^2$, you would get:

$f(x) = 3(x-2)^2 + 1$

Conclusion

In conclusion, function transformations are a powerful tool for creating new functions from existing ones. By understanding the different types of function transformations and how to apply them, you can create a wide range of functions to solve real-world problems.

Key Takeaways

• Function transformations are changes made to a function to create a new function.
• There are several types of function transformations, including vertical stretch, vertical compression, horizontal stretch, horizontal compression, and translation.
• The order of operations for function transformations is parentheses, exponents, multiplication and division, and addition and subtraction.
• To determine the resulting function after applying a sequence of transformations, you apply each transformation in the given order.

Further Reading

For more information on function transformations, please refer to the following resources:

• Khan Academy: Function Transformations
• Math Is Fun: Function Transformations
• Wolfram MathWorld: Function Transformations