Translation 3 Units Right and 1 Unit Up

Oct 13, 2025 by windailys.com 149 views

Understanding the Problem

We are given a function ${$ f(x) = \ln x $}$ and asked to write a rule for a composite function ${$ g(x) $}$ that represents a translation 3 units right and 1 unit up, followed by a horizontal stretch by a factor of the graph of ${$ f(x) = \ln x $}$. This problem requires us to apply the concepts of function composition, translations, and stretches to create a new function that represents the desired transformation.

Translation 3 Units Right and 1 Unit Up

To begin, we need to understand how to represent a translation 3 units right and 1 unit up. A translation 3 units right can be represented by replacing ${$ x $}$ with ${$ x - 3 $}$, and a translation 1 unit up can be represented by adding 1 to the function. Therefore, the translation of the function ${$ f(x) = \ln x $}$ 3 units right and 1 unit up can be represented as ${$ f(x - 3) + 1 $}$.

Horizontal Stretch by a Factor

Next, we need to apply a horizontal stretch by a factor of the graph of ${$ f(x) = \ln x $}$. A horizontal stretch by a factor of ${$ k $}$ can be represented by replacing ${$ x $}$ with ${$ kx $}$. In this case, we want to apply a horizontal stretch by a factor of ${$ k $}$, where ${$ k $}$ is a constant. Therefore, the horizontal stretch of the function ${$ f(x - 3) + 1 $}$ can be represented as ${$ f(k(x - 3)) + 1 $}$.

Combining the Transformations

Now, we need to combine the translations and the horizontal stretch to create the composite function ${$ g(x) $}$. We can do this by substituting the expression for the horizontal stretch into the expression for the translation: ${$ g(x) = f(k(x - 3)) + 1 $}$.

Simplifying the Composite Function

To simplify the composite function, we can substitute the expression for ${$ f(x) = \ln x $}$ into the expression for ${$ g(x) $}$: ${$ g(x) = \ln(k(x - 3)) + 1 $}$. This is the final rule for the composite function ${$ g(x) $}$ that represents a translation 3 units right and 1 unit up, followed by a horizontal stretch by a factor of the graph of ${$ f(x) = \ln x $}$.

Example and Graph

Let's consider an example where ${$ k = 2 $}$. In this case, the composite function ${$ g(x) $}$ becomes ${$ g(x) = \ln(2(x - 3)) + 1 $}$. The graph of this function can be obtained by applying the translations and the horizontal stretch to the graph of ${$ f(x) = \ln x $}$.

Conclusion

In this article, we have written a rule for a composite function ${$ g(x) $}$ that represents a translation 3 units right and 1 unit up, followed by a horizontal stretch by a factor of the graph of ${$ f(x) = \ln x $}$. We have applied the concepts of function composition, translations, and stretches to create a new function that represents the desired transformation. The final rule for the composite function is ${$ g(x) = \ln(k(x - 3)) + 1 $}$, where ${$ k $}$ is a constant.

Key Takeaways

To write a rule for a composite function, we need to apply the concepts of function composition, translations, and stretches.
A translation 3 units right can be represented by replacing ${$ x $}$ with ${$ x - 3 $}$, and a translation 1 unit up can be represented by adding 1 to the function.
A horizontal stretch by a factor of ${$ k $}$ can be represented by replacing ${$ x $}$ with ${$ kx $}$.
The composite function ${$ g(x) $}$ can be obtained by substituting the expression for the horizontal stretch into the expression for the translation.

Understanding the Problem

In our previous article, we wrote a rule for a composite function ${$ g(x) $}$ that represents a translation 3 units right and 1 unit up, followed by a horizontal stretch by a factor of the graph of ${$ f(x) = \ln x $}$. In this article, we will answer some frequently asked questions about writing a rule for a composite function.

Q: What is a composite function?

A: A composite function is a function that is obtained by applying one or more functions to the input of another function. In the case of the composite function ${$ g(x) $}$, we are applying a translation and a horizontal stretch to the function ${$ f(x) = \ln x $}$.

Q: How do I write a rule for a composite function?

A: To write a rule for a composite function, you need to apply the concepts of function composition, translations, and stretches. You can start by identifying the individual functions that make up the composite function, and then apply the necessary transformations to obtain the final rule.

Q: What is a translation?

A: A translation is a transformation that moves a graph a certain distance in the x-direction or y-direction. In the case of the composite function ${$ g(x) $}$, we are applying a translation 3 units right and 1 unit up.

Q: What is a horizontal stretch?

A: A horizontal stretch is a transformation that stretches a graph horizontally by a certain factor. In the case of the composite function ${$ g(x) $}$, we are applying a horizontal stretch by a factor of ${$ k $}$.

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

A: To apply a horizontal stretch to a function, you need to replace ${$ x $}$ with ${$ kx $}$, where ${$ k $}$ is the factor by which you want to stretch the graph.

Q: What is the final rule for the composite function ${$ g(x) $}$?

A: The final rule for the composite function ${$ g(x) $}$ is ${$ g(x) = \ln(k(x - 3)) + 1 $}$, where ${$ k $}$ is a constant.

Q: Can I apply multiple transformations to a function?

A: Yes, you can apply multiple transformations to a function. For example, you can apply a translation, a horizontal stretch, and a vertical stretch to a function.

Q: How do I apply multiple transformations to a function?

A: To apply multiple transformations to a function, you need to apply each transformation in the correct order. For example, if you want to apply a translation, a horizontal stretch, and a vertical stretch to a function, you need to apply the translation first, followed by the horizontal stretch, and finally the vertical stretch.

Q: What are some common transformations that can be applied to a function?

A: Some common transformations that can be applied to a function include translations, horizontal stretches, vertical stretches, reflections, and rotations.

Q: How do I determine the order in which to apply transformations to a function?

A: To determine the order in which to apply transformations to a function, you need to consider the type of transformation and the direction in which it is applied. For example, if you want to apply a translation and a horizontal stretch to a function, you need to apply the translation first, followed by the horizontal stretch.

Conclusion

In this article, we have answered some frequently asked questions about writing a rule for a composite function. We have covered topics such as composite functions, translations, horizontal stretches, and multiple transformations. By understanding these concepts, you can write a rule for a composite function that represents a complex transformation.

Key Takeaways

A composite function is a function that is obtained by applying one or more functions to the input of another function.
To write a rule for a composite function, you need to apply the concepts of function composition, translations, and stretches.
A translation is a transformation that moves a graph a certain distance in the x-direction or y-direction.
A horizontal stretch is a transformation that stretches a graph horizontally by a certain factor.
You can apply multiple transformations to a function by applying each transformation in the correct order.