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 downwards. The function can be rewritten as $g(x)=(0.5)^{x} \cdot (0.5)^3 - 4$, which simplifies to $g(x)=(0.5)^{x} \cdot 0.125 - 4$.

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 positive.
• Range: The range of the function is all real numbers, since the function can take on any value.
• Asymptote: The function has a horizontal asymptote at $y=-4$, since the function approaches this value as $x$ approaches infinity.
• Intercepts: The function has a y-intercept at $(0,-4)$, since the function passes through this point.
• End behavior: The function approaches the horizontal asymptote as $x$ approaches infinity, and it approaches the y-intercept as $x$ approaches negative infinity.

Graphing the Exponential Function

To graph the exponential function, we can use the following steps:

1. Plot the horizontal asymptote: Plot the horizontal line $y=-4$.
2. Plot the y-intercept: Plot the point $(0,-4)$.
3. Plot the exponential function: Plot the points $(x, (0.5)^{x} \cdot 0.125 - 4)$ for various values of $x$.
4. Connect the points: Connect the points to form a smooth curve.

Graphing the Function

The graph of the function $g(x)=(0.5)^{x+3}-4$ is a smooth curve that approaches the horizontal asymptote $y=-4$ as $x$ approaches infinity. The curve passes through the y-intercept $(0,-4)$ and approaches the y-intercept as $x$ approaches negative infinity.

Key Features of the Graph

The graph of the function has the following key features:

• Horizontal asymptote: The graph approaches the horizontal asymptote $y=-4$ as $x$ approaches infinity.
• Y-intercept: The graph passes through the y-intercept $(0,-4)$.
• End behavior: The graph approaches the y-intercept as $x$ approaches negative infinity.
• Domain: The graph is defined for all real numbers.
• Range: The graph is defined for all real numbers.

Conclusion

The graph of the function $g(x)=(0.5)^{x+3}-4$ is a smooth curve that approaches the horizontal asymptote $y=-4$ as $x$ approaches infinity. The curve passes through the y-intercept $(0,-4)$ and approaches the y-intercept as $x$ approaches negative infinity. The graph has a horizontal asymptote, a y-intercept, and end behavior that approaches the y-intercept as $x$ approaches negative infinity.

Graph of the Function

Here is a graph of the function $g(x)=(0.5)^{x+3}-4$:

# Graph of the Function g(x)=(0.5)^{x+3}-4

## Horizontal Asymptote

* The graph approaches the horizontal asymptote y=-4 as x approaches infinity.

## Y-Intercept

* The graph passes through the y-intercept (0,-4).

## End Behavior

* The graph approaches the y-intercept as x approaches negative infinity.

## Domain

* The graph is defined for all real numbers.

## Range

* The graph is defined for all real numbers.

## Graph

* The graph is a smooth curve that approaches the horizontal asymptote y=-4 as x approaches infinity.
* The graph passes through the y-intercept (0,-4).
* The graph approaches the y-intercept as x approaches negative infinity.

Final Thoughts

The graph of the function $g(x)=(0.5)^{x+3}-4$ is a smooth curve that approaches the horizontal asymptote $y=-4$ as $x$ approaches infinity. The curve passes through the y-intercept $(0,-4)$ and approaches the y-intercept as $x$ approaches negative infinity. The graph has a horizontal asymptote, a y-intercept, and end behavior that approaches the y-intercept as $x$ approaches negative infinity.

Frequently Asked Questions

Q: What is the domain of the function $g(x)=(0.5)^{x+3}-4$?

A: The domain of the function is all real numbers, since the base of the exponential function is positive.

Q: What is the range of the function $g(x)=(0.5)^{x+3}-4$?

A: The range of the function 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 is $y=-4$, since the function approaches this value 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 is $(0,-4)$, since the function passes through this point.

Q: How does the function $g(x)=(0.5)^{x+3}-4$ behave as $x$ approaches negative infinity?

A: The function approaches the y-intercept $(0,-4)$ as $x$ approaches negative infinity.

Q: How does the function $g(x)=(0.5)^{x+3}-4$ behave as $x$ approaches infinity?

A: The function approaches the horizontal asymptote $y=-4$ 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 is that it approaches the horizontal asymptote $y=-4$ as $x$ approaches infinity, and it passes through the y-intercept $(0,-4)$.

Q: How can I graph the function $g(x)=(0.5)^{x+3}-4$?

A: To graph the function, you can use the following steps:

1. Plot the horizontal asymptote $y=-4$.
2. Plot the y-intercept $(0,-4)$.
3. Plot the points $(x, (0.5)^{x} \cdot 0.125 - 4)$ for various values of $x$.
4. Connect the points to form a smooth curve.

Q: What is the significance of the graph of the function $g(x)=(0.5)^{x+3}-4$?

A: The graph of the function is significant because it shows the behavior of the function as $x$ approaches infinity and negative infinity. It also shows the horizontal asymptote and the y-intercept of the function.

Q: Can I use the graph of the function $g(x)=(0.5)^{x+3}-4$ to make predictions about the behavior of the function?

A: Yes, you can use the graph of the function to make predictions about the behavior of the function. For example, you can use the graph to determine the value of the function at a given point, or to determine the behavior of the function as $x$ approaches infinity or negative infinity.

Conclusion

The graph of the function $g(x)=(0.5)^{x+3}-4$ is a smooth curve that approaches the horizontal asymptote $y=-4$ as $x$ approaches infinity. The curve passes through the y-intercept $(0,-4)$ and approaches the y-intercept as $x$ approaches negative infinity. The graph has a horizontal asymptote, a y-intercept, and end behavior that approaches the y-intercept as $x$ approaches negative infinity.