Select All The Correct Answers.Consider The Parent Function $f(x) = E^x$ And The Transformed Function $g(x) = -f(x) - 4$. Which Features Of The Function Are Affected?- Range- Y Y Y -intercept- Domain- Horizontal Asymptote-

Introduction

In mathematics, transformations play a crucial role in understanding and analyzing functions. When a function is transformed, its features such as range, $y$-intercept, domain, and horizontal asymptote are affected. In this article, we will explore the effects of transformations on the parent function $f(x) = e^x$ and the transformed function $g(x) = -f(x) - 4$.

Understanding the Parent Function

The parent function $f(x) = e^x$ is an exponential function that has a base of $e$. This function has several key features that are essential to understanding its behavior.

• Range: The range of the function $f(x) = e^x$ is all positive real numbers, denoted as $(0, \infty)$.
• $y$-intercept: The $y$-intercept of the function $f(x) = e^x$ is 1, as $f(0) = e^0 = 1$.
• Domain: The domain of the function $f(x) = e^x$ is all real numbers, denoted as $(-\infty, \infty)$.
• Horizontal asymptote: The horizontal asymptote of the function $f(x) = e^x$ is $y = 0$, as the function approaches 0 as $x$ approaches negative infinity.

Understanding the Transformed Function

The transformed function $g(x) = -f(x) - 4$ is obtained by applying two transformations to the parent function $f(x) = e^x$.

• Reflection across the $x$-axis: The function $g(x) = -f(x)$ is obtained by reflecting the parent function $f(x) = e^x$ across the $x$-axis. This reflection affects the range and $y$-intercept of the function.
• Vertical shift: The function $g(x) = -f(x) - 4$ is obtained by shifting the reflected function $g(x) = -f(x)$ down by 4 units. This vertical shift affects the $y$-intercept of the function.

Effects of Transformations on the Function

The transformations applied to the parent function $f(x) = e^x$ affect the following features of the function:

• Range: The range of the transformed function $g(x) = -f(x) - 4$ is all negative real numbers, denoted as $(-\infty, 0)$.
• $y$-intercept: The $y$-intercept of the transformed function $g(x) = -f(x) - 4$ is -5, as $g(0) = -f(0) - 4 = -1 - 4 = -5$.
• Domain: The domain of the transformed function $g(x) = -f(x) - 4$ is all real numbers, denoted as $(-\infty, \infty)$.
• Horizontal asymptote: The horizontal asymptote of the transformed function $g(x) = -f(x) - 4$ is $y = -4$, as the function approaches -4 as $x$ approaches negative infinity.

Conclusion

In conclusion, the transformations applied to the parent function $f(x) = e^x$ affect the range, $y$-intercept, and horizontal asymptote of the function. The reflection across the $x$-axis and vertical shift affect the range and $y$-intercept of the function, while the horizontal asymptote is affected by the reflection across the $x$-axis.

Key Takeaways

• Reflection across the $x$-axis affects the range and $y$-intercept of the function.
• Vertical shift affects the $y$-intercept of the function.
• Horizontal asymptote is affected by reflection across the $x$-axis.

Practice Problems

1. What is the range of the transformed function $g(x) = -f(x) - 4$?
2. What is the $y$-intercept of the transformed function $g(x) = -f(x) - 4$?
3. What is the domain of the transformed function $g(x) = -f(x) - 4$?
4. What is the horizontal asymptote of the transformed function $g(x) = -f(x) - 4$?

Answer Key

1. The range of the transformed function $g(x) = -f(x) - 4$ is all negative real numbers, denoted as $(-\infty, 0)$.
2. The $y$-intercept of the transformed function $g(x) = -f(x) - 4$ is -5.
3. The domain of the transformed function $g(x) = -f(x) - 4$ is all real numbers, denoted as $(-\infty, \infty)$.
4. The horizontal asymptote of the transformed function $g(x) = -f(x) - 4$ is $y = -4$.
   Q&A: Transforming Functions =============================

Q1: What is the effect of reflecting a function across the x-axis?

A1: When a function is reflected across the x-axis, its range and y-intercept are affected. The range becomes the negative of the original range, and the y-intercept becomes the negative of the original y-intercept.

Q2: What is the effect of shifting a function vertically?

A2: When a function is shifted vertically, its y-intercept is affected. The y-intercept is shifted up or down by the amount of the shift.

Q3: What is the effect of reflecting a function across the x-axis and then shifting it vertically?

A3: When a function is reflected across the x-axis and then shifted vertically, its range, y-intercept, and horizontal asymptote are affected. The range becomes the negative of the original range, the y-intercept becomes the negative of the original y-intercept, and the horizontal asymptote is shifted vertically.

Q4: How do transformations affect the domain of a function?

A4: Transformations do not affect the domain of a function. The domain remains the same regardless of the transformations applied to the function.

Q5: What is the effect of reflecting a function across the x-axis on its horizontal asymptote?

A5: When a function is reflected across the x-axis, its horizontal asymptote is affected. The horizontal asymptote becomes the negative of the original horizontal asymptote.

Q6: What is the effect of shifting a function vertically on its horizontal asymptote?

A6: When a function is shifted vertically, its horizontal asymptote is not affected. The horizontal asymptote remains the same regardless of the vertical shift.

Q7: How do transformations affect the graph of a function?

A7: Transformations affect the graph of a function by changing its position, shape, and size. Reflections across the x-axis and y-axis change the orientation of the graph, while vertical and horizontal shifts change the position of the graph.

Q8: What is the difference between a reflection and a shift?

A8: A reflection is a transformation that changes the orientation of a graph, while a shift is a transformation that changes the position of a graph. Reflections are typically denoted by a negative sign, while shifts are denoted by a positive or negative value.

Q9: How do transformations affect the equation of a function?

A9: Transformations affect the equation of a function by changing its coefficients and constants. Reflections across the x-axis and y-axis change the sign of the coefficients, while vertical and horizontal shifts change the constants.

Q10: What is the importance of understanding transformations in mathematics?

A10: Understanding transformations is important in mathematics because it allows us to analyze and manipulate functions in a variety of ways. Transformations are used in a wide range of mathematical applications, including algebra, geometry, and calculus.

Conclusion

In conclusion, transformations are an essential part of mathematics, and understanding them is crucial for analyzing and manipulating functions. By understanding the effects of transformations on the range, y-intercept, domain, and horizontal asymptote of a function, we can better analyze and manipulate functions in a variety of mathematical applications.