Which Transformations Take The Graph Of F ( X ) = 5 X F(x)=5^x F ( X ) = 5 X To The Graph Of G ( X ) = − 5 X + 7 G(x)=-5^{x+7} G ( X ) = − 5 X + 7 ?Choose Each Correct Answer:A. The Graph Is Reflected Across The X X X -axis.B. The Graph Is Translated To The Left 7 Units.

Introduction

Exponential functions are a fundamental concept in mathematics, and understanding their transformations is crucial for solving various problems in calculus, algebra, and other branches of mathematics. In this article, we will explore the transformations that take the graph of $f(x)=5^x$ to the graph of $g(x)=-5^{x+7}$.

Understanding the Original Function

The original function is $f(x)=5^x$. This is an exponential function with base 5, and its graph is a curve that increases rapidly as $x$ increases. The graph of $f(x)$ has a horizontal asymptote at $y=0$ and a vertical asymptote at $x=-\infty$.

Understanding the Target Function

The target function is $g(x)=-5^{x+7}$. This function is also an exponential function with base 5, but it has a negative coefficient in front of the exponent. This means that the graph of $g(x)$ will be a reflection of the graph of $f(x)$ across the $x$-axis. Additionally, the graph of $g(x)$ will be shifted to the left by 7 units.

Reflection Across the $x$-Axis

The graph of $g(x)$ is a reflection of the graph of $f(x)$ across the $x$-axis. This means that for every point $(x, y)$ on the graph of $f(x)$, there is a corresponding point $(x, -y)$ on the graph of $g(x)$. This transformation is represented by the equation $g(x)=-f(x)$.

Translation to the Left

The graph of $g(x)$ is also translated to the left by 7 units. This means that for every point $(x, y)$ on the graph of $f(x)$, there is a corresponding point $(x-7, y)$ on the graph of $g(x)$. This transformation is represented by the equation $g(x)=f(x+7)$.

Combining Reflection and Translation

To combine the reflection and translation transformations, we can use the equation $g(x)=-f(x+7)$. This equation represents the graph of $g(x)$ as a reflection of the graph of $f(x)$ across the $x$-axis, followed by a translation to the left by 7 units.

Conclusion

In conclusion, the graph of $g(x)=-5^{x+7}$ can be obtained from the graph of $f(x)=5^x$ by reflecting it across the $x$-axis and translating it to the left by 7 units. This can be represented by the equation $g(x)=-f(x+7)$. Understanding these transformations is crucial for solving various problems in mathematics and other branches of science.

Discussion

Which transformations take the graph of $f(x)=5^x$ to the graph of $g(x)=-5^{x+7}$?

A. The graph is reflected across the $x$-axis. B. The graph is translated to the left 7 units.

The correct answer is:

A. The graph is reflected across the $x$-axis. B. The graph is translated to the left 7 units.

Q&A: Transformations of Exponential Functions

Q: What is the effect of reflecting the graph of $f(x)=5^x$ across the $x$-axis?

A: Reflecting the graph of $f(x)=5^x$ across the $x$-axis results in a new function $g(x)=-f(x)$. This means that for every point $(x, y)$ on the graph of $f(x)$, there is a corresponding point $(x, -y)$ on the graph of $g(x)$.

Q: What is the effect of translating the graph of $f(x)=5^x$ to the left by 7 units?

A: Translating the graph of $f(x)=5^x$ to the left by 7 units results in a new function $g(x)=f(x+7)$. This means that for every point $(x, y)$ on the graph of $f(x)$, there is a corresponding point $(x-7, y)$ on the graph of $g(x)$.

Q: How do you combine reflection and translation transformations?

A: To combine reflection and translation transformations, you can use the equation $g(x)=-f(x+7)$. This equation represents the graph of $g(x)$ as a reflection of the graph of $f(x)$ across the $x$-axis, followed by a translation to the left by 7 units.

Q: What is the effect of reflecting the graph of $f(x)=5^x$ across the $y$-axis?

A: Reflecting the graph of $f(x)=5^x$ across the $y$-axis results in a new function $g(x)=f(-x)$. This means that for every point $(x, y)$ on the graph of $f(x)$, there is a corresponding point $(-x, y)$ on the graph of $g(x)$.

Q: What is the effect of stretching the graph of $f(x)=5^x$ vertically by a factor of 2?

A: Stretching the graph of $f(x)=5^x$ vertically by a factor of 2 results in a new function $g(x)=2f(x)$. This means that for every point $(x, y)$ on the graph of $f(x)$, there is a corresponding point $(x, 2y)$ on the graph of $g(x)$.

Q: What is the effect of compressing the graph of $f(x)=5^x$ horizontally by a factor of 2?

A: Compressing the graph of $f(x)=5^x$ horizontally by a factor of 2 results in a new function $g(x)=f(2x)$. This means that for every point $(x, y)$ on the graph of $f(x)$, there is a corresponding point $(\frac{x}{2}, y)$ on the graph of $g(x)$.

Q: How do you determine the equation of a transformed function?

A: To determine the equation of a transformed function, you need to identify the type of transformation and the value of the transformation parameter. For example, if the graph of $f(x)$ is reflected across the $x$-axis and translated to the left by 7 units, the equation of the transformed function is $g(x)=-f(x+7)$.

Conclusion

In conclusion, understanding the transformations of exponential functions is crucial for solving various problems in mathematics and other branches of science. By mastering the concepts of reflection, translation, stretching, and compressing, you can analyze and solve complex problems involving exponential functions.