How Does The Graph Of $y=-5e^x$ Compare To The Graph Of The Parent Function $f(x)=e^x$?A. It Is Reflected Across The $y$-axis And Stretched Vertically By A Factor Of 5.B. It Is Reflected Across The $y$-axis And

Introduction

The exponential function $f(x) = e^x$ is a fundamental function in mathematics, and its graph is a key concept in understanding various mathematical and real-world phenomena. When we modify the function by multiplying it by a constant, we can observe changes in its graph. In this article, we will explore how the graph of $y = -5e^x$ compares to the graph of the parent function $f(x) = e^x$.

Understanding the Parent Function

The parent function $f(x) = e^x$ is an exponential function that has a base of $e$, where $e$ is a mathematical constant approximately equal to 2.71828. The graph of $f(x) = e^x$ is a continuous, increasing curve that passes through the point (0, 1). The graph has a horizontal asymptote at $y = 0$ and a vertical asymptote at $x = -\infty$.

Modifying the Function

When we modify the function $f(x) = e^x$ by multiplying it by a constant, we get the function $y = -5e^x$. The constant $-5$ is a negative number, which means that the graph of $y = -5e^x$ will be reflected across the $x$-axis compared to the graph of $f(x) = e^x$.

Comparing the Graphs

To compare the graphs of $y = -5e^x$ and $f(x) = e^x$, let's consider the following key features:

• Reflection across the $y$-axis: The graph of $y = -5e^x$ is reflected across the $y$-axis compared to the graph of $f(x) = e^x$. This means that if we replace $x$ with $-x$ in the equation $y = -5e^x$, we get the equation $y = -5e^{-x}$, which is the equation of the graph of $f(x) = e^x$.
• Vertical stretch: The graph of $y = -5e^x$ is stretched vertically by a factor of 5 compared to the graph of $f(x) = e^x$. This means that if we multiply the $y$-coordinate of any point on the graph of $f(x) = e^x$ by 5, we get the $y$-coordinate of the corresponding point on the graph of $y = -5e^x$.

Conclusion

In conclusion, the graph of $y = -5e^x$ is reflected across the $y$-axis and stretched vertically by a factor of 5 compared to the graph of the parent function $f(x) = e^x$. This means that if we replace $x$ with $-x$ in the equation $y = -5e^x$, we get the equation $y = -5e^{-x}$, which is the equation of the graph of $f(x) = e^x$. The vertical stretch factor of 5 means that if we multiply the $y$-coordinate of any point on the graph of $f(x) = e^x$ by 5, we get the $y$-coordinate of the corresponding point on the graph of $y = -5e^x$.

Graphical Representation

The following graph shows the comparison between the graphs of $y = -5e^x$ and $f(x) = e^x$.

![](graph.png)

Key Takeaways

• The graph of $y = -5e^x$ is reflected across the $y$-axis compared to the graph of $f(x) = e^x$.
• The graph of $y = -5e^x$ is stretched vertically by a factor of 5 compared to the graph of $f(x) = e^x$.
• If we replace $x$ with $-x$ in the equation $y = -5e^x$, we get the equation $y = -5e^{-x}$, which is the equation of the graph of $f(x) = e^x$.

Real-World Applications

The comparison between the graphs of $y = -5e^x$ and $f(x) = e^x$ has various real-world applications, such as:

• Population growth: The graph of $y = -5e^x$ can be used to model the population growth of a species that is declining exponentially.
• Chemical reactions: The graph of $y = -5e^x$ can be used to model the concentration of a chemical species that is decreasing exponentially over time.
• Financial modeling: The graph of $y = -5e^x$ can be used to model the value of an investment that is decreasing exponentially over time.

Conclusion

Frequently Asked Questions

Q1: What is the parent function $f(x) = e^x$?

A1: The parent function $f(x) = e^x$ is an exponential function that has a base of $e$, where $e$ is a mathematical constant approximately equal to 2.71828. The graph of $f(x) = e^x$ is a continuous, increasing curve that passes through the point (0, 1).

Q2: How does the graph of $y = -5e^x$ compare to the graph of $f(x) = e^x$?

A2: The graph of $y = -5e^x$ is reflected across the $y$-axis and stretched vertically by a factor of 5 compared to the graph of $f(x) = e^x$.

Q3: What is the effect of multiplying the function $f(x) = e^x$ by a negative constant on its graph?

A3: Multiplying the function $f(x) = e^x$ by a negative constant reflects the graph across the $y$-axis.

Q4: What is the effect of multiplying the function $f(x) = e^x$ by a positive constant greater than 1 on its graph?

A4: Multiplying the function $f(x) = e^x$ by a positive constant greater than 1 stretches the graph vertically.

Q5: How can we determine the equation of the graph of $y = -5e^x$ if we know the equation of the graph of $f(x) = e^x$?

A5: We can determine the equation of the graph of $y = -5e^x$ by replacing $x$ with $-x$ in the equation of the graph of $f(x) = e^x$ and multiplying the result by $-5$.

Q6: What are some real-world applications of the comparison between the graphs of $y = -5e^x$ and $f(x) = e^x$?

A6: Some real-world applications of the comparison between the graphs of $y = -5e^x$ and $f(x) = e^x$ include population growth, chemical reactions, and financial modeling.

Q7: How can we use the graph of $y = -5e^x$ to model population growth?

A7: We can use the graph of $y = -5e^x$ to model population growth by assuming that the population is declining exponentially over time.

Q8: How can we use the graph of $y = -5e^x$ to model chemical reactions?

A8: We can use the graph of $y = -5e^x$ to model chemical reactions by assuming that the concentration of a chemical species is decreasing exponentially over time.

Q9: How can we use the graph of $y = -5e^x$ to model financial investments?

A9: We can use the graph of $y = -5e^x$ to model financial investments by assuming that the value of an investment is decreasing exponentially over time.

Q10: What are some key takeaways from the comparison between the graphs of $y = -5e^x$ and $f(x) = e^x$?

A10: Some key takeaways from the comparison between the graphs of $y = -5e^x$ and $f(x) = e^x$ include the reflection of the graph across the $y$-axis and the vertical stretch by a factor of 5.

Conclusion

In conclusion, the comparison between the graphs of $y = -5e^x$ and $f(x) = e^x$ has various real-world applications, such as population growth, chemical reactions, and financial modeling. By understanding the key features of the graph of $y = -5e^x$, we can use it to model various phenomena in different fields.