Which Of The Following Is The Graph Of $f(x) = -0.5|x+3| - 2$?

Introduction

In mathematics, a piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. The graph of a piecewise function can be a combination of different types of graphs, such as linear, quadratic, or absolute value functions. In this article, we will explore the graph of the function $f(x) = -0.5|x+3| - 2$.

The Function

The given function is $f(x) = -0.5|x+3| - 2$. This function is a piecewise function, where the absolute value function $|x+3|$ is defined as:

$$|x+3| = \begin{cases} x+3, & \text{if } x \geq -3 \\ -(x+3), & \text{if } x < -3 \end{cases}$$

Graphing the Function

To graph the function $f(x) = -0.5|x+3| - 2$, we need to consider two cases: when $x \geq -3$ and when $x < -3$.

Case 1: $x \geq -3$

When $x \geq -3$, the absolute value function $|x+3|$ is equal to $x+3$. Therefore, the function $f(x) = -0.5|x+3| - 2$ becomes:

$$f(x) = -0.5(x+3) - 2$$

This is a linear function with a slope of $-0.5$ and a y-intercept of $-2.5$. The graph of this function is a line with a negative slope, passing through the point $(-3, -5)$.

Case 2: $x < -3$

When $x < -3$, the absolute value function $|x+3|$ is equal to $-(x+3)$. Therefore, the function $f(x) = -0.5|x+3| - 2$ becomes:

$$f(x) = -0.5(-(x+3)) - 2$$

This is also a linear function with a slope of $0.5$ and a y-intercept of $-2.5$. The graph of this function is a line with a positive slope, passing through the point $(-3, -5)$.

Combining the Graphs

To graph the function $f(x) = -0.5|x+3| - 2$, we need to combine the graphs of the two cases. The graph of the function is a V-shaped graph, with a vertex at the point $(-3, -5)$. The graph has a negative slope for $x \geq -3$ and a positive slope for $x < -3$.

Conclusion

In conclusion, the graph of the function $f(x) = -0.5|x+3| - 2$ is a V-shaped graph with a vertex at the point $(-3, -5)$. The graph has a negative slope for $x \geq -3$ and a positive slope for $x < -3$. This graph is a combination of two linear functions, each defined on a specific interval of the domain.

Key Takeaways

• The graph of a piecewise function can be a combination of different types of graphs.
• The graph of the function $f(x) = -0.5|x+3| - 2$ is a V-shaped graph with a vertex at the point $(-3, -5)$.
• The graph has a negative slope for $x \geq -3$ and a positive slope for $x < -3$.

Real-World Applications

The graph of a piecewise function has many real-world applications, such as:

• Modeling the behavior of a physical system that changes its behavior at a specific point.
• Representing the cost or revenue of a business that changes its pricing strategy at a specific point.
• Modeling the growth or decline of a population that changes its growth rate at a specific point.

Final Thoughts

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Introduction

In our previous article, we explored the graph of the function $f(x) = -0.5|x+3| - 2$. We discussed how the graph is a combination of two linear functions, each defined on a specific interval of the domain. In this article, we will answer some frequently asked questions about the graph of a piecewise function.

Q: What is a piecewise function?

A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. The graph of a piecewise function can be a combination of different types of graphs, such as linear, quadratic, or absolute value functions.

Q: How do I graph a piecewise function?

To graph a piecewise function, you need to consider each sub-function separately. You need to identify the intervals of the domain where each sub-function is defined, and then graph each sub-function on its respective interval.

Q: What is the vertex of a piecewise function?

The vertex of a piecewise function is the point where the function changes its behavior. In the case of the function $f(x) = -0.5|x+3| - 2$, the vertex is the point $(-3, -5)$.

Q: How do I determine the slope of a piecewise function?

To determine the slope of a piecewise function, you need to consider each sub-function separately. You need to identify the slope of each sub-function, and then determine the slope of the overall function.

Q: Can a piecewise function have a negative slope?

Yes, a piecewise function can have a negative slope. In the case of the function $f(x) = -0.5|x+3| - 2$, the slope is negative for $x \geq -3$.

Q: Can a piecewise function have a positive slope?

Yes, a piecewise function can have a positive slope. In the case of the function $f(x) = -0.5|x+3| - 2$, the slope is positive for $x < -3$.

Q: How do I use a piecewise function in real-world applications?

A piecewise function can be used to model the behavior of a physical system that changes its behavior at a specific point. It can also be used to represent the cost or revenue of a business that changes its pricing strategy at a specific point.

Q: What are some common types of piecewise functions?

Some common types of piecewise functions include:

• Linear piecewise functions
• Quadratic piecewise functions
• Absolute value piecewise functions
• Polynomial piecewise functions

Q: How do I graph a piecewise function with multiple sub-functions?

To graph a piecewise function with multiple sub-functions, you need to consider each sub-function separately. You need to identify the intervals of the domain where each sub-function is defined, and then graph each sub-function on its respective interval.

Conclusion

In conclusion, the graph of a piecewise function is a combination of different types of graphs, such as linear, quadratic, or absolute value functions. Understanding the graph of a piecewise function is essential in many fields of study, including mathematics, physics, and engineering. By answering these frequently asked questions, we hope to have provided a better understanding of the graph of a piecewise function.

Key Takeaways

• A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.
• The graph of a piecewise function can be a combination of different types of graphs.
• The vertex of a piecewise function is the point where the function changes its behavior.
• A piecewise function can have a negative or positive slope.
• A piecewise function can be used to model the behavior of a physical system or represent the cost or revenue of a business.

Real-World Applications

The graph of a piecewise function has many real-world applications, such as:

• Modeling the behavior of a physical system that changes its behavior at a specific point.
• Representing the cost or revenue of a business that changes its pricing strategy at a specific point.
• Modeling the growth or decline of a population that changes its growth rate at a specific point.

Final Thoughts

In conclusion, the graph of a piecewise function is a powerful tool for modeling and analyzing complex systems. By understanding the graph of a piecewise function, we can gain insights into the behavior of physical systems, businesses, and populations.