Which Translation Takes The First Function F F F To The Second G G G ? F ( X ) = 3 X G ( X ) = 3 X + 5 F(x) = 3^x \quad G(x) = 3^x + 5 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 Translated Left 5 Units. D.

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Which Translation Takes the First Function ff to the Second gg?

Understanding Function Transformations

In mathematics, function transformations are essential concepts that help us understand how functions change under various operations. When we have two functions, f(x)f(x) and g(x)g(x), and we want to find the translation that takes f(x)f(x) to g(x)g(x), we need to analyze the differences between the two functions.

The Given Functions

Let's consider the two functions:

f(x)=3xf(x) = 3^x

g(x)=3x+5g(x) = 3^x + 5

Analyzing the Functions

To determine the translation that takes f(x)f(x) to g(x)g(x), we need to compare the two functions. The main difference between the two functions is the constant term. The function f(x)f(x) has no constant term, while the function g(x)g(x) has a constant term of 55.

Translation Up 5 Units

If we translate the graph of f(x)f(x) up 5 units, we get a new function:

f(x)+5=3x+5f(x) + 5 = 3^x + 5

This new function is identical to the function g(x)g(x). Therefore, the graph of f(x)f(x) translated up 5 units is the graph of g(x)g(x).

Conclusion

Based on our analysis, we can conclude that the graph of f(x)f(x) translated up 5 units is the graph of g(x)g(x). Therefore, the correct answer is:

A. The graph is translated up 5 units.

Why Not Down 5 Units?

Some of you might be wondering why we didn't consider the option of translating the graph of f(x)f(x) down 5 units. The reason is that translating the graph of f(x)f(x) down 5 units would result in a function:

f(x)−5=3x−5f(x) - 5 = 3^x - 5

This function is not identical to the function g(x)g(x). Therefore, translating the graph of f(x)f(x) down 5 units is not the correct answer.

Why Not Left 5 Units?

Another option that might seem plausible is translating the graph of f(x)f(x) left 5 units. However, this would result in a function:

f(x+5)=3x+5f(x + 5) = 3^{x+5}

This function is not identical to the function g(x)g(x). Therefore, translating the graph of f(x)f(x) left 5 units is not the correct answer.

Summary

In summary, we have analyzed the two functions f(x)=3xf(x) = 3^x and g(x)=3x+5g(x) = 3^x + 5 and determined that the graph of f(x)f(x) translated up 5 units is the graph of g(x)g(x). Therefore, the correct answer is:

A. The graph is translated up 5 units.

Key Takeaways

  • Function transformations are essential concepts in mathematics.
  • To determine the translation that takes one function to another, we need to analyze the differences between the two functions.
  • Translating the graph of f(x)f(x) up 5 units results in the graph of g(x)g(x).
  • Translating the graph of f(x)f(x) down 5 units or left 5 units does not result in the graph of g(x)g(x).

Further Reading

If you want to learn more about function transformations, I recommend checking out the following resources:

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

Practice Problems

Try solving the following practice problems to test your understanding of function transformations:

  1. Find the translation that takes the function f(x)=2xf(x) = 2^x to the function g(x)=2x+3g(x) = 2^x + 3.
  2. Find the translation that takes the function f(x)=x2f(x) = x^2 to the function g(x)=x2+2g(x) = x^2 + 2.
  3. Find the translation that takes the function f(x)=sin⁡(x)f(x) = \sin(x) to the function g(x)=sin⁡(x)+1g(x) = \sin(x) + 1.

Conclusion

In conclusion, we have analyzed the two functions f(x)=3xf(x) = 3^x and g(x)=3x+5g(x) = 3^x + 5 and determined that the graph of f(x)f(x) translated up 5 units is the graph of g(x)g(x). We have also discussed why translating the graph of f(x)f(x) down 5 units or left 5 units does not result in the graph of g(x)g(x). I hope this article has helped you understand function transformations better.
Q&A: Function Transformations

Understanding Function Transformations

In our previous article, we discussed how to determine the translation that takes one function to another. We analyzed the functions f(x)=3xf(x) = 3^x and g(x)=3x+5g(x) = 3^x + 5 and determined that the graph of f(x)f(x) translated up 5 units is the graph of g(x)g(x). 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 in a new function. Function transformations can be vertical (up or down) or horizontal (left or right).

Q: What are the different types of function transformations?

A: There are four main types of function transformations:

  1. Vertical Stretch: This is a transformation that stretches the graph of a function vertically.
  2. Vertical Compression: This is a transformation that compresses the graph of a function vertically.
  3. Horizontal Stretch: This is a transformation that stretches the graph of a function horizontally.
  4. Horizontal Compression: This is a transformation that compresses the graph of a function horizontally.

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

A: To determine the type of function transformation, you need to analyze the differences between the two functions. Look for changes in the constant term, coefficient, or base of the function.

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

A: A vertical transformation changes the height of the graph, while a horizontal transformation changes the width of the graph.

Q: Can a function have multiple transformations?

A: Yes, a function can have multiple transformations. For example, a function can be translated up 5 units and then stretched vertically by a factor of 2.

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

A: To apply a function transformation to a function, you need to follow these steps:

  1. Identify the type of transformation.
  2. Determine the amount of the transformation (e.g., 5 units up).
  3. Apply the transformation to the function.

Q: What are some common function transformations?

A: Some common function transformations include:

  1. Translation: This is a transformation that moves the graph of a function up or down or left or right.
  2. Dilation: This is a transformation that stretches or compresses the graph of a function vertically or horizontally.
  3. Reflection: This is a transformation that flips the graph of a function over a line or a point.

Q: How do I graph a function with multiple transformations?

A: To graph a function with multiple transformations, you need to apply each transformation in order. For example, if you want to graph a function that is translated up 5 units and then stretched vertically by a factor of 2, you would first translate the graph up 5 units and then stretch it vertically by a factor of 2.

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

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

  1. Physics: Function transformations are used to model the motion of objects.
  2. Engineering: Function transformations are used to design and optimize systems.
  3. Economics: Function transformations are used to model economic systems and make predictions.

Conclusion

In conclusion, function transformations are an essential concept in mathematics that have many real-world applications. By understanding function transformations, you can analyze and solve problems in a variety of fields. We hope this article has helped you understand function transformations better.

Practice Problems

Try solving the following practice problems to test your understanding of function transformations:

  1. Find the translation that takes the function f(x)=2xf(x) = 2^x to the function g(x)=2x+3g(x) = 2^x + 3.
  2. Find the dilation that takes the function f(x)=x2f(x) = x^2 to the function g(x)=2x2g(x) = 2x^2.
  3. Find the reflection that takes the function f(x)=sin⁡(x)f(x) = \sin(x) to the function g(x)=−sin⁡(x)g(x) = -\sin(x).

Further Reading

If you want to learn more about function transformations, we recommend checking out the following resources:

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

Key Takeaways

  • Function transformations are a change in the graph of a function that results in a new function.
  • There are four main types of function transformations: vertical stretch, vertical compression, horizontal stretch, and horizontal compression.
  • Function transformations have many real-world applications, including physics, engineering, and economics.