Describe The Transformation That Maps The Translations From The Parent Function, $f(x)=2^x$, To The Function $g(x)=2^{x+7}$. State Any Changes To The Domain, Range, Intercepts, Asymptotes, And End Behavior.

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Introduction

In mathematics, transformations are essential concepts that help us understand how functions change under various operations. In this article, we will explore the transformation that maps the parent function f(x)=2xf(x)=2^x to the function g(x)=2x+7g(x)=2^{x+7}. We will examine the changes to the domain, range, intercepts, asymptotes, and end behavior of the function.

Parent Function: f(x)=2xf(x)=2^x

The parent function f(x)=2xf(x)=2^x is an exponential function with a base of 2. This function has a characteristic "S" shape, where the value of the function increases rapidly as xx increases. The domain of f(x)f(x) is all real numbers, and the range is all positive real numbers.

Transformation: g(x)=2x+7g(x)=2^{x+7}

To transform the parent function f(x)=2xf(x)=2^x into g(x)=2x+7g(x)=2^{x+7}, we need to apply a horizontal shift of 7 units to the right. This means that for every value of xx in f(x)f(x), we add 7 to get the corresponding value of xx in g(x)g(x).

Changes to the Domain

The domain of g(x)g(x) is also all real numbers, just like the domain of f(x)f(x). However, the transformation does not change the domain of the function.

Changes to the Range

The range of g(x)g(x) is also all positive real numbers, just like the range of f(x)f(x). However, the transformation does not change the range of the function.

Changes to the Intercepts

The yy-intercept of f(x)f(x) is (0,1)(0, 1), since f(0)=20=1f(0)=2^0=1. The yy-intercept of g(x)g(x) is also (0,27)=(0,128)(0, 2^7)= (0, 128), since g(0)=20+7=27=128g(0)=2^{0+7}=2^7=128.

Changes to the Asymptotes

The horizontal asymptote of f(x)f(x) is the xx-axis, since as xx approaches infinity, f(x)f(x) approaches infinity. The horizontal asymptote of g(x)g(x) is also the xx-axis, since as xx approaches infinity, g(x)g(x) approaches infinity.

Changes to the End Behavior

The end behavior of f(x)f(x) is that as xx approaches negative infinity, f(x)f(x) approaches 0, and as xx approaches positive infinity, f(x)f(x) approaches infinity. The end behavior of g(x)g(x) is also that as xx approaches negative infinity, g(x)g(x) approaches 0, and as xx approaches positive infinity, g(x)g(x) approaches infinity.

Conclusion

In conclusion, the transformation that maps the parent function f(x)=2xf(x)=2^x to the function g(x)=2x+7g(x)=2^{x+7} is a horizontal shift of 7 units to the right. This transformation does not change the domain, range, intercepts, asymptotes, or end behavior of the function.

Key Takeaways

  • The transformation g(x)=2x+7g(x)=2^{x+7} is a horizontal shift of 7 units to the right of the parent function f(x)=2xf(x)=2^x.
  • The domain, range, intercepts, asymptotes, and end behavior of the function remain unchanged under this transformation.

References

Q: What is the transformation that maps the parent function f(x)=2xf(x)=2^x to the function g(x)=2x+7g(x)=2^{x+7}?

A: The transformation that maps the parent function f(x)=2xf(x)=2^x to the function g(x)=2x+7g(x)=2^{x+7} is a horizontal shift of 7 units to the right.

Q: How does the transformation affect the domain of the function?

A: The transformation does not change the domain of the function. The domain of g(x)g(x) is also all real numbers, just like the domain of f(x)f(x).

Q: How does the transformation affect the range of the function?

A: The transformation does not change the range of the function. The range of g(x)g(x) is also all positive real numbers, just like the range of f(x)f(x).

Q: What is the effect of the transformation on the intercepts of the function?

A: The yy-intercept of f(x)f(x) is (0,1)(0, 1), since f(0)=20=1f(0)=2^0=1. The yy-intercept of g(x)g(x) is also (0,27)=(0,128)(0, 2^7)= (0, 128), since g(0)=20+7=27=128g(0)=2^{0+7}=2^7=128.

Q: How does the transformation affect the asymptotes of the function?

A: The horizontal asymptote of f(x)f(x) is the xx-axis, since as xx approaches infinity, f(x)f(x) approaches infinity. The horizontal asymptote of g(x)g(x) is also the xx-axis, since as xx approaches infinity, g(x)g(x) approaches infinity.

Q: What is the effect of the transformation on the end behavior of the function?

A: The end behavior of f(x)f(x) is that as xx approaches negative infinity, f(x)f(x) approaches 0, and as xx approaches positive infinity, f(x)f(x) approaches infinity. The end behavior of g(x)g(x) is also that as xx approaches negative infinity, g(x)g(x) approaches 0, and as xx approaches positive infinity, g(x)g(x) approaches infinity.

Q: Can you provide an example of how to apply the transformation to a specific function?

A: Yes, let's consider the function f(x)=2xf(x)=2^x. To apply the transformation to get g(x)=2x+7g(x)=2^{x+7}, we need to add 7 to the input of the function. For example, if we want to find the value of g(2)g(2), we need to add 7 to the input, so g(2)=22+7=29=512g(2)=2^{2+7}=2^9=512.

Q: Are there any other transformations that can be applied to the function?

A: Yes, there are other transformations that can be applied to the function, such as vertical shifts, horizontal stretches, and reflections. However, in this article, we have only discussed the horizontal shift transformation.

Q: Can you provide a summary of the key takeaways from this article?

A: Yes, the key takeaways from this article are:

  • The transformation g(x)=2x+7g(x)=2^{x+7} is a horizontal shift of 7 units to the right of the parent function f(x)=2xf(x)=2^x.
  • The domain, range, intercepts, asymptotes, and end behavior of the function remain unchanged under this transformation.

References