Use What You Know About Translations Of Functions To Analyze The Graph Of The Function $f(x)=(0.5)^{x-5}+8$. You May Wish To Graph It And Its Parent Function, $y=0.5^x$, On The Same Axes.1. The Parent Function $y=0.5^x$ Is

Feb 26, 2025 by ADMIN 223 views

**Analyzing the Graph of a Function through Translation**

Understanding the Parent Function

The parent function $y=0.5^x$ is a basic exponential function with a base of 0.5. This function has a limited range and domain, and its graph is a decreasing curve that approaches the x-axis as x increases. The graph of the parent function can be used as a reference to analyze the graph of the given function $f(x)=(0.5)^{x-5}+8$ .

Translation of Functions

When analyzing the graph of a function, it is essential to understand the concept of translation. Translation refers to the process of shifting the graph of a function up, down, left, or right, or a combination of these movements. In the given function $f(x)=(0.5)^{x-5}+8$ , we can identify two types of translations: horizontal and vertical.

Horizontal Translation

The horizontal translation in the given function is represented by the term $x-5$ inside the exponent. This means that the graph of the function is shifted 5 units to the right compared to the graph of the parent function $y=0.5^x$ . As a result, the x-intercept of the graph of $f(x)$ is 5 units to the right of the x-intercept of the graph of $y=0.5^x$ .

Vertical Translation

The vertical translation in the given function is represented by the term $+8$ added to the function. This means that the graph of the function is shifted 8 units upwards compared to the graph of the parent function $y=0.5^x$ . As a result, the y-intercept of the graph of $f(x)$ is 8 units above the y-intercept of the graph of $y=0.5^x$ .

Graphing the Functions

To visualize the graph of the given function $f(x)=(0.5)^{x-5}+8$ , it is helpful to graph the parent function $y=0.5^x$ on the same axes. By comparing the two graphs, we can see the effect of the horizontal and vertical translations on the graph of the function.

Key Features of the Graph

The graph of the function $f(x)=(0.5)^{x-5}+8$ has several key features that can be identified by analyzing the graph of the parent function $y=0.5^x$ . Some of these features include:

X-intercept: The x-intercept of the graph of $f(x)$ is 5 units to the right of the x-intercept of the graph of $y=0.5^x$ .
Y-intercept: The y-intercept of the graph of $f(x)$ is 8 units above the y-intercept of the graph of $y=0.5^x$ .
Asymptote: The graph of $f(x)$ has a horizontal asymptote at y=8, which is the result of the vertical translation.
Domain and Range: The domain and range of the graph of $f(x)$ are affected by the horizontal translation, but the graph still approaches the x-axis as x increases.

Conclusion

In conclusion, analyzing the graph of a function through translation is a powerful tool for understanding the behavior of the function. By identifying the horizontal and vertical translations in the given function $f(x)=(0.5)^{x-5}+8$ , we can determine the key features of the graph, including the x-intercept, y-intercept, asymptote, and domain and range. By graphing the parent function $y=0.5^x$ on the same axes, we can visualize the effect of the translations on the graph of the function.

Graphing the Parent Function

To graph the parent function $y=0.5^x$ , we can use a graphing calculator or a computer algebra system. The graph of the parent function is a decreasing curve that approaches the x-axis as x increases.

Graphing the Given Function

To graph the given function $f(x)=(0.5)^{x-5}+8$ , we can use a graphing calculator or a computer algebra system. The graph of the given function is a decreasing curve that approaches the x-axis as x increases, but it is shifted 5 units to the right and 8 units upwards compared to the graph of the parent function.

Key Takeaways

The graph of the function $f(x)=(0.5)^{x-5}+8$ can be analyzed through translation.
The horizontal translation in the given function is represented by the term $x-5$ inside the exponent.
The vertical translation in the given function is represented by the term $+8$ added to the function.
The graph of the given function has several key features, including the x-intercept, y-intercept, asymptote, and domain and range.

Final Thoughts

Q: What is the parent function in the given function $f(x)=(0.5)^{x-5}+8$ ?

A: The parent function in the given function is $y=0.5^x$ . This is a basic exponential function with a base of 0.5.

Q: What is the effect of the horizontal translation in the given function?

A: The horizontal translation in the given function is represented by the term $x-5$ inside the exponent. This means that the graph of the function is shifted 5 units to the right compared to the graph of the parent function $y=0.5^x$ .

Q: What is the effect of the vertical translation in the given function?

A: The vertical translation in the given function is represented by the term $+8$ added to the function. This means that the graph of the function is shifted 8 units upwards compared to the graph of the parent function $y=0.5^x$ .

Q: How can we visualize the graph of the given function?

A: We can visualize the graph of the given function by graphing the parent function $y=0.5^x$ on the same axes. By comparing the two graphs, we can see the effect of the horizontal and vertical translations on the graph of the function.

Q: What are some key features of the graph of the given function?

A: Some key features of the graph of the given function include:

X-intercept: The x-intercept of the graph of $f(x)$ is 5 units to the right of the x-intercept of the graph of $y=0.5^x$ .
Y-intercept: The y-intercept of the graph of $f(x)$ is 8 units above the y-intercept of the graph of $y=0.5^x$ .
Asymptote: The graph of $f(x)$ has a horizontal asymptote at y=8, which is the result of the vertical translation.
Domain and Range: The domain and range of the graph of $f(x)$ are affected by the horizontal translation, but the graph still approaches the x-axis as x increases.

Q: How can we use the graph of the parent function to analyze the graph of the given function?

A: We can use the graph of the parent function to analyze the graph of the given function by identifying the horizontal and vertical translations. By comparing the two graphs, we can see the effect of the translations on the graph of the function.

Q: What are some real-world applications of analyzing the graph of a function through translation?

A: Analyzing the graph of a function through translation has several real-world applications, including:

Modeling population growth: By analyzing the graph of a function through translation, we can model population growth and understand how different factors affect population size.
Analyzing economic data: By analyzing the graph of a function through translation, we can understand how different economic factors affect economic growth and stability.
Understanding chemical reactions: By analyzing the graph of a function through translation, we can understand how different chemical reactions affect the rate of reaction and the amount of product formed.

Q: How can we use technology to analyze the graph of a function through translation?

A: We can use technology, such as graphing calculators or computer algebra systems, to analyze the graph of a function through translation. These tools allow us to visualize the graph of the function and identify the horizontal and vertical translations.

Q: What are some common mistakes to avoid when analyzing the graph of a function through translation?

A: Some common mistakes to avoid when analyzing the graph of a function through translation include:

Failing to identify the horizontal and vertical translations: It is essential to identify the horizontal and vertical translations in the function to understand the behavior of the graph.
Not using technology to visualize the graph: Using technology to visualize the graph of the function can help us identify the horizontal and vertical translations and understand the behavior of the graph.
Not considering the domain and range of the function: The domain and range of the function can affect the behavior of the graph, and it is essential to consider these factors when analyzing the graph through translation.