The Graph Of $f(x)=-\frac{1}{2}\left(\frac{1}{3}\right)^{x-4}+5$ Is Transformed As Follows: - Shifted Downward 5 Units- Shifted Left 4 Units- Stretched Vertically By A Factor Of 8- Reflected About The $x$-axisWhat Is The Equation Of

Introduction

In mathematics, the graph of a function is a visual representation of the relationship between the input and output values of the function. When a function is transformed, its graph changes in a specific way. In this article, we will explore the transformations of the graph of the function $f(x)=-\frac{1}{2}\left(\frac{1}{3}\right)^{x-4}+5$ and determine the equation of the transformed graph.

Original Function

The original function is given by the equation $f(x)=-\frac{1}{2}\left(\frac{1}{3}\right)^{x-4}+5$. This is an exponential function with a base of $\frac{1}{3}$ and a vertical shift of 5 units.

Exponential Function

The exponential function is defined as $f(x)=a\cdot b^{x-c}+d$, where $a$, $b$, $c$, and $d$ are constants. In this case, $a=-\frac{1}{2}$, $b=\frac{1}{3}$, $c=4$, and $d=5$.

Graph of the Original Function

The graph of the original function is a decreasing exponential curve that passes through the point $(4, 5)$. The graph has a vertical asymptote at $x=4$ and a horizontal asymptote at $y=0$.

Transformations

The graph of the original function is transformed in the following ways:

1. Shifted downward 5 units: The graph is shifted downward by 5 units, which means that the new function is $f(x)=-\frac{1}{2}\left(\frac{1}{3}\right)^{x-4}$.
2. Shifted left 4 units: The graph is shifted left by 4 units, which means that the new function is $f(x)=-\frac{1}{2}\left(\frac{1}{3}\right)^{x}$.
3. Stretched vertically by a factor of 8: The graph is stretched vertically by a factor of 8, which means that the new function is $f(x)=-4\left(\frac{1}{3}\right)^{x}$.
4. Reflected about the $x$-axis: The graph is reflected about the $x$-axis, which means that the new function is $f(x)=4\left(\frac{1}{3}\right)^{x}$.

Equation of the Transformed Function

The equation of the transformed function is $f(x)=4\left(\frac{1}{3}\right)^{x}$. This is an exponential function with a base of $\frac{1}{3}$ and a vertical stretch of 4 units.

Graph of the Transformed Function

The graph of the transformed function is an increasing exponential curve that passes through the point $(0, 4)$. The graph has a vertical asymptote at $x=-\infty$ and a horizontal asymptote at $y=0$.

Conclusion

In conclusion, the graph of the function $f(x)=-\frac{1}{2}\left(\frac{1}{3}\right)^{x-4}+5$ is transformed in the following ways:

• Shifted downward 5 units
• Shifted left 4 units
• Stretched vertically by a factor of 8
• Reflected about the $x$-axis

The equation of the transformed function is $f(x)=4\left(\frac{1}{3}\right)^{x}$. This is an exponential function with a base of $\frac{1}{3}$ and a vertical stretch of 4 units.

References

• [1] "Exponential Functions". Math Open Reference. Retrieved 2023-02-27.
• [2] "Graphing Exponential Functions". Purplemath. Retrieved 2023-02-27.

Additional Resources

• [1] Khan Academy. "Exponential Functions". Retrieved 2023-02-27.
• [2] Mathway. "Exponential Functions". Retrieved 2023-02-27.
  The Graph of a Function: Transformations and Reflections - Q&A ===========================================================

Introduction

In our previous article, we explored the transformations of the graph of the function $f(x)=-\frac{1}{2}\left(\frac{1}{3}\right)^{x-4}+5$ and determined the equation of the transformed graph. In this article, we will answer some frequently asked questions about the graph of a function and its transformations.

Q&A

Q: What is the difference between a shift and a stretch?

A: A shift is a change in the position of the graph, while a stretch is a change in the size of the graph. For example, shifting a graph left by 4 units means that the graph is moved 4 units to the left, while stretching a graph vertically by a factor of 8 means that the graph is enlarged by a factor of 8 in the vertical direction.

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

A: To determine the equation of a transformed function, you need to apply the transformations to the original function. For example, if the original function is $f(x)=a\cdot b^{x-c}+d$ and the transformations are a shift left by 4 units, a shift downward by 5 units, and a vertical stretch by a factor of 8, the equation of the transformed function is $f(x)=8a\cdot b^{x-4}-5$.

Q: What is the difference between a reflection and a rotation?

A: A reflection is a change in the orientation of the graph, while a rotation is a change in the position of the graph. For example, reflecting a graph about the $x$-axis means that the graph is flipped upside down, while rotating a graph by 90 degrees means that the graph is rotated by 90 degrees clockwise or counterclockwise.

Q: How do I graph a transformed function?

A: To graph a transformed function, you need to apply the transformations to the original graph. For example, if the original graph is a decreasing exponential curve and the transformations are a shift left by 4 units, a shift downward by 5 units, and a vertical stretch by a factor of 8, the graph of the transformed function is an increasing exponential curve that passes through the point $(0, 8)$.

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

A: A vertical asymptote is a line that the graph approaches as the input value increases or decreases without bound, while a horizontal asymptote is a line that the graph approaches as the input value increases or decreases without bound. For example, the graph of the function $f(x)=\frac{1}{x}$ has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.

Q: How do I determine the equation of a function with multiple transformations?

A: To determine the equation of a function with multiple transformations, you need to apply the transformations in the correct order. For example, if the original function is $f(x)=a\cdot b^{x-c}+d$ and the transformations are a shift left by 4 units, a shift downward by 5 units, a vertical stretch by a factor of 8, and a reflection about the $x$-axis, the equation of the transformed function is $f(x)=-8a\cdot b^{x-4}-5$.

Conclusion

In conclusion, the graph of a function can be transformed in various ways, including shifts, stretches, reflections, and rotations. By understanding the effects of these transformations, you can determine the equation of a transformed function and graph it accurately.

References

• [1] "Exponential Functions". Math Open Reference. Retrieved 2023-02-27.
• [2] "Graphing Exponential Functions". Purplemath. Retrieved 2023-02-27.

Additional Resources

• [1] Khan Academy. "Exponential Functions". Retrieved 2023-02-27.
• [2] Mathway. "Exponential Functions". Retrieved 2023-02-27.