Write A Rule For $g$ That Represents A Translation 4 Units Right And 1 Unit Down, Followed By A Vertical Shrink By A Factor Of $\frac{1}{3}$ Of The Graph Of \$f(x)=e^{-x}$[/tex\].$g(x)=$ $\square$

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Understanding the Problem

To create a rule for $g$ that represents a translation 4 units right and 1 unit down, followed by a vertical shrink by a factor of $\frac{1}{3}$ of the graph of $f(x)=e^{-x}$, we need to understand the individual transformations involved.

Translation 4 Units Right and 1 Unit Down

The translation of a function $f(x)$ by $a$ units to the right and $b$ units down is given by the rule $f(x-a) - b$. In this case, we want to translate the graph of $f(x)=e^{-x}$ 4 units to the right and 1 unit down. This can be achieved by replacing $x$ with $x-4$ and subtracting 1 from the function.

Vertical Shrink by a Factor of $\frac{1}{3}$

A vertical shrink by a factor of $k$ of a function $f(x)$ is given by the rule $kf(x)$. In this case, we want to shrink the graph of $f(x)=e^{-x}$ vertically by a factor of $\frac{1}{3}$.

Composing the Transformations

To find the rule for $g$, we need to compose the translation and vertical shrink transformations. First, we apply the translation transformation to the function $f(x)=e^{-x}$, which gives us $f(x-4) - 1$. Then, we apply the vertical shrink transformation to the result, which gives us $\frac{1}{3}(f(x-4) - 1)$.

Simplifying the Expression

To simplify the expression, we can start by evaluating the inner function $f(x-4)$, which is $e^{-(x-4)}$. Then, we can substitute this expression into the outer function, which gives us $\frac{1}{3}(e^{-(x-4)} - 1)$.

Final Rule for $g$

The final rule for $g$ is $g(x) = \frac{1}{3}(e^{-(x-4)} - 1)$. This rule represents a translation 4 units right and 1 unit down, followed by a vertical shrink by a factor of $\frac{1}{3}$ of the graph of $f(x)=e^{-x}$.

Example Use Case

To illustrate the use of the rule for $g$, let's consider an example. Suppose we want to find the value of $g(2)$. We can start by evaluating the inner function $f(x-4)$ at $x=2$, which gives us $f(2-4) = f(-2) = e^2$. Then, we can substitute this expression into the outer function, which gives us $g(2) = \frac{1}{3}(e^2 - 1)$.

Conclusion

In this article, we have derived a rule for $g$ that represents a translation 4 units right and 1 unit down, followed by a vertical shrink by a factor of $\frac{1}{3}$ of the graph of $f(x)=e^{-x}$. The final rule for $g$ is $g(x) = \frac{1}{3}(e^{-(x-4)} - 1)$. We have also provided an example use case to illustrate the use of the rule for $g$.

Key Takeaways

  • To create a rule for $g$ that represents a translation 4 units right and 1 unit down, followed by a vertical shrink by a factor of $\frac{1}{3}$ of the graph of $f(x)=e^{-x}$, we need to understand the individual transformations involved.
  • The translation of a function $f(x)$ by $a$ units to the right and $b$ units down is given by the rule $f(x-a) - b$.
  • A vertical shrink by a factor of $k$ of a function $f(x)$ is given by the rule $kf(x)$.
  • To find the rule for $g$, we need to compose the translation and vertical shrink transformations.
  • The final rule for $g$ is $g(x) = \frac{1}{3}(e^{-(x-4)} - 1)$.

Further Reading

For further reading on transformations and function composition, we recommend the following resources:

  • Khan Academy: Transformations of Functions
  • Math Is Fun: Function Composition
  • Wolfram MathWorld: Function Composition

By following the steps outlined in this article, you can create a rule for $g$ that represents a translation 4 units right and 1 unit down, followed by a vertical shrink by a factor of $\frac{1}{3}$ of the graph of $f(x)=e^{-x}$.

Understanding the Problem

In the previous article, we derived a rule for $g$ that represents a translation 4 units right and 1 unit down, followed by a vertical shrink by a factor of $\frac{1}{3}$ of the graph of $f(x)=e^{-x}$. In this article, we will answer some frequently asked questions about composing transformations.

Q: What is the difference between a horizontal and vertical translation?

A: A horizontal translation is a transformation that shifts the graph of a function to the left or right, while a vertical translation is a transformation that shifts the graph of a function up or down.

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

A: To apply a horizontal translation to a function, you need to replace $x$ with $x-a$, where $a$ is the number of units you want to translate the graph to the right or left.

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

A: To apply a vertical translation to a function, you need to add or subtract a constant value from the function.

Q: What is the difference between a horizontal and vertical shrink?

A: A horizontal shrink is a transformation that compresses the graph of a function horizontally, while a vertical shrink is a transformation that compresses the graph of a function vertically.

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

A: To apply a horizontal shrink to a function, you need to replace $x$ with $\frac{x}{k}$, where $k$ is the scale factor.

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

A: To apply a vertical shrink to a function, you need to multiply the function by a constant value.

Q: Can I apply multiple transformations to a function?

A: Yes, you can apply multiple transformations to a function. To do this, you need to compose the transformations in the correct order.

Q: How do I compose multiple transformations?

A: To compose multiple transformations, you need to apply each transformation in the correct order. For example, if you want to apply a horizontal translation, followed by a vertical shrink, you need to first replace $x$ with $x-a$, and then multiply the function by a constant value.

Q: What is the final rule for $g$?

A: The final rule for $g$ is $g(x) = \frac{1}{3}(e^{-(x-4)} - 1)$.

Q: Can I use the rule for $g$ to find the value of $g(2)$?

A: Yes, you can use the rule for $g$ to find the value of $g(2)$. To do this, you need to substitute $x=2$ into the rule for $g$.

Q: What is the value of $g(2)$?

A: The value of $g(2)$ is $\frac{1}{3}(e^2 - 1)$.

Conclusion

In this article, we have answered some frequently asked questions about composing transformations. We have also provided examples and explanations to help you understand the concepts.

Key Takeaways

  • A horizontal translation is a transformation that shifts the graph of a function to the left or right.
  • A vertical translation is a transformation that shifts the graph of a function up or down.
  • A horizontal shrink is a transformation that compresses the graph of a function horizontally.
  • A vertical shrink is a transformation that compresses the graph of a function vertically.
  • You can apply multiple transformations to a function by composing them in the correct order.

Further Reading

For further reading on transformations and function composition, we recommend the following resources:

  • Khan Academy: Transformations of Functions
  • Math Is Fun: Function Composition
  • Wolfram MathWorld: Function Composition

By following the steps outlined in this article, you can answer questions about composing transformations and apply multiple transformations to a function.