Key Features of the Parent Function

Oct 13, 2025 by magicalceylontours.com 145 views

Understanding the Parent Function

The parent function of $f(x)=(0.5)^{x-5}+8$ is $y=0.5^x$ . This function is an exponential function with a base of 0.5, which means that as $x$ increases, the value of $y$ decreases. The graph of this function is a decreasing exponential curve that passes through the point $(0, 1)$ .

Key Features of the Parent Function

Domain and Range: The domain of the parent function is all real numbers, and the range is all positive real numbers.
Asymptotes: The parent function has a horizontal asymptote at $y=0$ .
Intercepts: The parent function has a $y$ -intercept at $(0, 1)$ .
End Behavior: As $x$ approaches infinity, $y$ approaches 0. As $x$ approaches negative infinity, $y$ approaches infinity.

Graphing the Parent Function

To graph the parent function, we can use a graphing calculator or software. The graph of the parent function is a decreasing exponential curve that passes through the point $(0, 1)$ . The graph has a horizontal asymptote at $y=0$ and a $y$ -intercept at $(0, 1)$ .

Analyzing the Function $f(x)=(0.5)^{x-5}+8$

The function $f(x)=(0.5)^{x-5}+8$ is a transformation of the parent function $y=0.5^x$ . The transformation involves a horizontal shift of 5 units to the right and a vertical shift of 8 units up.

Key Features of the Function $f(x)=(0.5)^{x-5}+8$

Domain and Range: The domain of the function is all real numbers, and the range is all positive real numbers greater than or equal to 8.
Asymptotes: The function has a horizontal asymptote at $y=8$ .
Intercepts: The function has a $y$ -intercept at $(0, 9)$ .
End Behavior: As $x$ approaches infinity, $y$ approaches 8. As $x$ approaches negative infinity, $y$ approaches infinity.

Graphing the Function $f(x)=(0.5)^{x-5}+8$

To graph the function, we can use a graphing calculator or software. The graph of the function is a decreasing exponential curve that passes through the point $(0, 9)$ . The graph has a horizontal asymptote at $y=8$ and a $y$ -intercept at $(0, 9)$ .

Comparing the Graphs of the Parent Function and the Function $f(x)=(0.5)^{x-5}+8$

The graph of the function $f(x)=(0.5)^{x-5}+8$ is similar to the graph of the parent function $y=0.5^x$ , but it is shifted 5 units to the right and 8 units up. The graph of the function has a horizontal asymptote at $y=8$ , which is 8 units above the horizontal asymptote of the parent function.

Conclusion

In conclusion, the function $f(x)=(0.5)^{x-5}+8$ is a transformation of the parent function $y=0.5^x$ . The transformation involves a horizontal shift of 5 units to the right and a vertical shift of 8 units up. The graph of the function is a decreasing exponential curve that passes through the point $(0, 9)$ . The graph has a horizontal asymptote at $y=8$ and a $y$ -intercept at $(0, 9)$ .

Frequently Asked Questions

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

A: The parent function of $f(x)=(0.5)^{x-5}+8$ is $y=0.5^x$ . This function is an exponential function with a base of 0.5, which means that as $x$ increases, the value of $y$ decreases.

Q: What are the key features of the parent function $y=0.5^x$ ?

A: The key features of the parent function $y=0.5^x$ are:

Domain and Range: The domain of the parent function is all real numbers, and the range is all positive real numbers.
Asymptotes: The parent function has a horizontal asymptote at $y=0$ .
Intercepts: The parent function has a $y$ -intercept at $(0, 1)$ .
End Behavior: As $x$ approaches infinity, $y$ approaches 0. As $x$ approaches negative infinity, $y$ approaches infinity.

Q: What is the effect of the transformation $(x-5)$ on the parent function $y=0.5^x$ ?

A: The transformation $(x-5)$ shifts the parent function $y=0.5^x$ 5 units to the right. This means that the new function $f(x)=(0.5)^{x-5}+8$ has a horizontal asymptote at $y=8$ and a $y$ -intercept at $(0, 9)$ .

Q: What is the effect of the transformation $+8$ on the parent function $y=0.5^x$ ?

A: The transformation $+8$ shifts the parent function $y=0.5^x$ 8 units up. This means that the new function $f(x)=(0.5)^{x-5}+8$ has a horizontal asymptote at $y=8$ and a $y$ -intercept at $(0, 9)$ .

Q: How do the graphs of the parent function and the function $f(x)=(0.5)^{x-5}+8$ compare?

A: The graph of the function $f(x)=(0.5)^{x-5}+8$ is similar to the graph of the parent function $y=0.5^x$ , but it is shifted 5 units to the right and 8 units up. The graph of the function has a horizontal asymptote at $y=8$ , which is 8 units above the horizontal asymptote of the parent function.

Q: What are the key features of the function $f(x)=(0.5)^{x-5}+8$ ?

A: The key features of the function $f(x)=(0.5)^{x-5}+8$ are:

Domain and Range: The domain of the function is all real numbers, and the range is all positive real numbers greater than or equal to 8.
Asymptotes: The function has a horizontal asymptote at $y=8$ .
Intercepts: The function has a $y$ -intercept at $(0, 9)$ .
End Behavior: As $x$ approaches infinity, $y$ approaches 8. As $x$ approaches negative infinity, $y$ approaches infinity.

Q: How can I graph the function $f(x)=(0.5)^{x-5}+8$ ?

A: You can graph the function $f(x)=(0.5)^{x-5}+8$ using a graphing calculator or software. The graph of the function is a decreasing exponential curve that passes through the point $(0, 9)$ . The graph has a horizontal asymptote at $y=8$ and a $y$ -intercept at $(0, 9)$ .

Q: What is the significance of the horizontal asymptote of the function $f(x)=(0.5)^{x-5}+8$ ?

A: The horizontal asymptote of the function $f(x)=(0.5)^{x-5}+8$ is $y=8$ . This means that as $x$ approaches infinity, $y$ approaches 8. This is an important feature of the function, as it indicates the behavior of the function as $x$ becomes very large.