Which Translation Takes The First Function $f$ To The Second $g$?Given:${ F(x) = 3^x }$ { G(x) = 3^x + 5 \} A. The Graph Is Translated Up 5 Units.B. The Graph Is Translated Down 5 Units.C. The Graph Is

Which Translation Takes the First Function $f$ to the Second $g$?

Understanding Function Transformations

In mathematics, function transformations are essential concepts that help us understand how functions change under various operations. When we are given two functions, $f(x)$ and $g(x)$, we need to determine the type of transformation that takes $f(x)$ to $g(x)$. In this article, we will explore the concept of function transformations and determine which translation takes the first function $f$ to the second $g$.

Given Functions

We are given two functions:

${ f(x) = 3^x }$

${ g(x) = 3^x + 5 }$

Understanding the Functions

The first function, $f(x) = 3^x$, is an exponential function with base 3. This function represents a curve that increases rapidly as $x$ increases.

The second function, $g(x) = 3^x + 5$, is also an exponential function with base 3, but it has been shifted upward by 5 units. This means that the graph of $g(x)$ is the same as the graph of $f(x)$, but shifted upward by 5 units.

Determining the Translation

To determine which translation takes the first function $f$ to the second function $g$, we need to analyze the difference between the two functions.

${ g(x) - f(x) = (3^x + 5) - 3^x = 5 }$

This shows that the difference between the two functions is a constant value of 5. This means that the graph of $g(x)$ is the same as the graph of $f(x)$, but shifted upward by 5 units.

Conclusion

Based on our analysis, we can conclude that the graph of $g(x)$ is the same as the graph of $f(x)$, but shifted upward by 5 units. Therefore, the correct answer is:

A. The graph is translated up 5 units.

Why is this the correct answer?

This is the correct answer because the graph of $g(x)$ is the same as the graph of $f(x)$, but shifted upward by 5 units. This means that the graph of $g(x)$ is a vertical translation of the graph of $f(x)$ by 5 units.

What is a vertical translation?

A vertical translation is a transformation that shifts a graph upward or downward by a certain distance. In this case, the graph of $g(x)$ is shifted upward by 5 units, which means that it is a vertical translation of the graph of $f(x)$ by 5 units.

Why is this important?

Understanding function transformations is essential in mathematics, as it helps us analyze and solve problems involving functions. By understanding how functions change under various operations, we can solve problems involving function transformations, which is a critical skill in mathematics.

Real-World Applications

Function transformations have many real-world applications, including:

• Physics: Function transformations are used to model real-world phenomena, such as the motion of objects under the influence of gravity.
• Engineering: Function transformations are used to design and analyze systems, such as electronic circuits and mechanical systems.
• Economics: Function transformations are used to model economic systems and make predictions about future economic trends.

Conclusion

In conclusion, we have determined that the graph of $g(x)$ is the same as the graph of $f(x)$, but shifted upward by 5 units. This means that the correct answer is:

A. The graph is translated up 5 units.

We hope that this article has helped you understand function transformations and how they are used to analyze and solve problems involving functions.
Q&A: Function Transformations

Understanding Function Transformations

In our previous article, we explored the concept of function transformations and determined which translation takes the first function $f$ to the second function $g$. In this article, we will answer some frequently asked questions about function transformations.

Q: What is a function transformation?

A: A function transformation is a change in the graph of a function that results from a change in the function's equation. This can include changes in the function's shape, size, or position.

Q: What are the different types of function transformations?

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

• Vertical translations: Shifting the graph of a function up or down by a certain distance.
• Horizontal translations: Shifting the graph of a function left or right by a certain distance.
• Stretches and compressions: Changing the scale of the graph of a function.
• Reflections: Flipping the graph of a function over a certain line or axis.

Q: How do I determine the type of function transformation?

A: To determine the type of function transformation, you need to analyze the difference between the two functions. This can be done by subtracting the first function from the second function.

Q: What is a vertical translation?

A: A vertical translation is a transformation that shifts a graph upward or downward by a certain distance. This can be represented by the equation $y = f(x) + c$, where $c$ is the vertical translation.

Q: What is a horizontal translation?

A: A horizontal translation is a transformation that shifts a graph left or right by a certain distance. This can be represented by the equation $y = f(x - c)$, where $c$ is the horizontal translation.

Q: How do I graph a function transformation?

A: To graph a function transformation, you need to graph the original function and then apply the transformation to the graph. This can be done by shifting the graph up or down, left or right, or by changing the scale of the graph.

Q: What are some real-world applications of function transformations?

A: Function transformations have many real-world applications, including:

• Physics: Function transformations are used to model real-world phenomena, such as the motion of objects under the influence of gravity.
• Engineering: Function transformations are used to design and analyze systems, such as electronic circuits and mechanical systems.
• Economics: Function transformations are used to model economic systems and make predictions about future economic trends.

Q: How do I determine the type of function transformation in a real-world problem?

A: To determine the type of function transformation in a real-world problem, you need to analyze the problem and identify the type of transformation that is being applied. This can be done by looking at the equation of the function and identifying the type of transformation that is being applied.

Q: What are some common mistakes to avoid when working with function transformations?

A: Some common mistakes to avoid when working with function transformations include:

• Not analyzing the equation of the function: Failing to analyze the equation of the function can lead to incorrect conclusions about the type of function transformation.
• Not identifying the type of transformation: Failing to identify the type of transformation can lead to incorrect conclusions about the function.
• Not graphing the function: Failing to graph the function can lead to incorrect conclusions about the function.

Conclusion

In conclusion, we have answered some frequently asked questions about function transformations. We hope that this article has helped you understand function transformations and how they are used to analyze and solve problems involving functions.