Which Is The Graph Of $g(x)=(0.5)^{x+3}-4$?
Understanding the Function
The given function is g(x) = (0.5)^(x+3) - 4. This is an exponential function with a base of 0.5, and it has been shifted vertically by 4 units. The function can be rewritten as g(x) = (0.5)^x * (0.5)^3 - 4, which simplifies to g(x) = (0.5)^x * 0.125 - 4.
Graphing the Function
To graph the function, we need to consider the following key features:
- Domain: The domain of the function is all real numbers, since the base of the exponential function is between 0 and 1.
- Range: The range of the function is all real numbers, since the function can take on any value.
- Asymptotes: The function has a horizontal asymptote at y = -4, since the function approaches -4 as x approaches infinity.
- Intercepts: The function has a y-intercept at (0, -4), since the function passes through the point (0, -4).
- End behavior: The function approaches -4 as x approaches infinity, and it approaches -4 as x approaches negative infinity.
Graphing the Exponential Function
The graph of the exponential function g(x) = (0.5)^x is a decreasing curve that approaches the x-axis as x approaches infinity. The graph of the function g(x) = (0.5)^(x+3) - 4 is a vertical shift of the graph of the exponential function, shifted down by 4 units.
Graphing the Vertical Shift
The vertical shift of the graph of the exponential function is a translation of the graph down by 4 units. This means that the graph of the function g(x) = (0.5)^(x+3) - 4 is the same as the graph of the exponential function, shifted down by 4 units.
Graphing the Function
The graph of the function g(x) = (0.5)^(x+3) - 4 is a decreasing curve that approaches the x-axis as x approaches infinity. The graph has a horizontal asymptote at y = -4, and it has a y-intercept at (0, -4). The graph approaches -4 as x approaches negative infinity.
Key Features of the Graph
The graph of the function g(x) = (0.5)^(x+3) - 4 has the following key features:
- Domain: All real numbers
- Range: All real numbers
- Asymptotes: Horizontal asymptote at y = -4
- Intercepts: Y-intercept at (0, -4)
- End behavior: Approaches -4 as x approaches infinity, and approaches -4 as x approaches negative infinity
Conclusion
The graph of the function g(x) = (0.5)^(x+3) - 4 is a decreasing curve that approaches the x-axis as x approaches infinity. The graph has a horizontal asymptote at y = -4, and it has a y-intercept at (0, -4). The graph approaches -4 as x approaches negative infinity.
Graph of the Function
The graph of the function g(x) = (0.5)^(x+3) - 4 is shown below:
[Insert graph of the function]
Key Takeaways
- The graph of the function g(x) = (0.5)^(x+3) - 4 is a decreasing curve that approaches the x-axis as x approaches infinity.
- The graph has a horizontal asymptote at y = -4, and it has a y-intercept at (0, -4).
- The graph approaches -4 as x approaches negative infinity.
Final Thoughts
The graph of the function g(x) = (0.5)^(x+3) - 4 is a complex function that requires careful analysis to understand its key features. The graph has a horizontal asymptote at y = -4, and it has a y-intercept at (0, -4). The graph approaches -4 as x approaches negative infinity.
Q: What is the domain of the function g(x) = (0.5)^(x+3) - 4?
A: The domain of the function g(x) = (0.5)^(x+3) - 4 is all real numbers, since the base of the exponential function is between 0 and 1.
Q: What is the range of the function g(x) = (0.5)^(x+3) - 4?
A: The range of the function g(x) = (0.5)^(x+3) - 4 is all real numbers, since the function can take on any value.
Q: What is the horizontal asymptote of the function g(x) = (0.5)^(x+3) - 4?
A: The horizontal asymptote of the function g(x) = (0.5)^(x+3) - 4 is y = -4, since the function approaches -4 as x approaches infinity.
Q: What is the y-intercept of the function g(x) = (0.5)^(x+3) - 4?
A: The y-intercept of the function g(x) = (0.5)^(x+3) - 4 is (0, -4), since the function passes through the point (0, -4).
Q: How does the graph of the function g(x) = (0.5)^(x+3) - 4 behave as x approaches negative infinity?
A: The graph of the function g(x) = (0.5)^(x+3) - 4 approaches -4 as x approaches negative infinity.
Q: How does the graph of the function g(x) = (0.5)^(x+3) - 4 behave as x approaches infinity?
A: The graph of the function g(x) = (0.5)^(x+3) - 4 approaches -4 as x approaches infinity.
Q: What is the key feature of the graph of the function g(x) = (0.5)^(x+3) - 4?
A: The key feature of the graph of the function g(x) = (0.5)^(x+3) - 4 is that it is a decreasing curve that approaches the x-axis as x approaches infinity.
Q: How can I graph the function g(x) = (0.5)^(x+3) - 4?
A: You can graph the function g(x) = (0.5)^(x+3) - 4 by using a graphing calculator or by plotting points on a coordinate plane.
Q: What is the significance of the graph of the function g(x) = (0.5)^(x+3) - 4?
A: The graph of the function g(x) = (0.5)^(x+3) - 4 is significant because it shows the behavior of the function as x approaches positive and negative infinity.
Q: Can I use the graph of the function g(x) = (0.5)^(x+3) - 4 to solve equations?
A: Yes, you can use the graph of the function g(x) = (0.5)^(x+3) - 4 to solve equations by finding the x-intercepts of the graph.
Q: How can I use the graph of the function g(x) = (0.5)^(x+3) - 4 to model real-world situations?
A: You can use the graph of the function g(x) = (0.5)^(x+3) - 4 to model real-world situations such as population growth, chemical reactions, and financial investments.
Q: What are some common applications of the function g(x) = (0.5)^(x+3) - 4?
A: Some common applications of the function g(x) = (0.5)^(x+3) - 4 include modeling population growth, chemical reactions, and financial investments.
Q: Can I use the graph of the function g(x) = (0.5)^(x+3) - 4 to make predictions about future events?
A: Yes, you can use the graph of the function g(x) = (0.5)^(x+3) - 4 to make predictions about future events by analyzing the behavior of the function as x approaches positive and negative infinity.
Q: How can I use the graph of the function g(x) = (0.5)^(x+3) - 4 to make informed decisions?
A: You can use the graph of the function g(x) = (0.5)^(x+3) - 4 to make informed decisions by analyzing the behavior of the function as x approaches positive and negative infinity and by using the graph to model real-world situations.
Q: What are some common mistakes to avoid when graphing the function g(x) = (0.5)^(x+3) - 4?
A: Some common mistakes to avoid when graphing the function g(x) = (0.5)^(x+3) - 4 include:
- Not using a graphing calculator or plotting points on a coordinate plane
- Not analyzing the behavior of the function as x approaches positive and negative infinity
- Not using the graph to model real-world situations
- Not making informed decisions based on the graph
Q: How can I improve my graphing skills when it comes to the function g(x) = (0.5)^(x+3) - 4?
A: You can improve your graphing skills when it comes to the function g(x) = (0.5)^(x+3) - 4 by:
- Practicing graphing the function using a graphing calculator or by plotting points on a coordinate plane
- Analyzing the behavior of the function as x approaches positive and negative infinity
- Using the graph to model real-world situations
- Making informed decisions based on the graph
Q: What are some resources that I can use to learn more about the graph of the function g(x) = (0.5)^(x+3) - 4?
A: Some resources that you can use to learn more about the graph of the function g(x) = (0.5)^(x+3) - 4 include:
- Graphing calculators
- Online graphing tools
- Math textbooks
- Online resources such as Khan Academy and Mathway
- Tutoring services
Q: Can I use the graph of the function g(x) = (0.5)^(x+3) - 4 to solve systems of equations?
A: Yes, you can use the graph of the function g(x) = (0.5)^(x+3) - 4 to solve systems of equations by finding the intersection points of the graph with other graphs.
Q: How can I use the graph of the function g(x) = (0.5)^(x+3) - 4 to model systems of equations?
A: You can use the graph of the function g(x) = (0.5)^(x+3) - 4 to model systems of equations by analyzing the behavior of the function as x approaches positive and negative infinity and by using the graph to model real-world situations.
Q: What are some common applications of the graph of the function g(x) = (0.5)^(x+3) - 4 in systems of equations?
A: Some common applications of the graph of the function g(x) = (0.5)^(x+3) - 4 in systems of equations include modeling population growth, chemical reactions, and financial investments.
Q: Can I use the graph of the function g(x) = (0.5)^(x+3) - 4 to make predictions about future events in systems of equations?
A: Yes, you can use the graph of the function g(x) = (0.5)^(x+3) - 4 to make predictions about future events in systems of equations by analyzing the behavior of the function as x approaches positive and negative infinity.
Q: How can I use the graph of the function g(x) = (0.5)^(x+3) - 4 to make informed decisions in systems of equations?
A: You can use the graph of the function g(x) = (0.5)^(x+3) - 4 to make informed decisions in systems of equations by analyzing the behavior of the function as x approaches positive and negative infinity and by using the graph to model real-world situations.
Q: What are some common mistakes to avoid when using the graph of the function g(x) = (0.5)^(x+3) - 4 in systems of equations?
A: Some common mistakes to avoid when using the graph of the function g(x) = (0.5)^(x+3) - 4 in systems of equations include:
- Not analyzing the behavior of the function as x approaches positive and negative infinity
- Not using the graph to model real-world situations
- Not making informed decisions based on the graph
- Not using the graph to solve systems of equations