Watch The Video And Then Solve The Problem Given Below.Graph The Function:$\[ F(x) = \begin{cases} \frac{1}{4}x, & \text{for } X \ \textless \ 0 \\ x + 2, & \text{for } X \geq 0 \end{cases} \\]Choose The Correct Graph Below.A. B. C. D.

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. In this article, we will explore how to graph a piecewise function, using the given function $f(x) = \begin{cases} \frac{1}{4}x, & \text{for } x \ \textless \ 0 \\ x + 2, & \text{for } x \geq 0 \end{cases}$ as an example.

Understanding the Piecewise Function

Before we can graph the function, we need to understand its behavior on different intervals of the domain. The given function has two sub-functions:

• For $x < 0$, the function is defined as $f(x) = \frac{1}{4}x$.
• For $x \geq 0$, the function is defined as $f(x) = x + 2$.

Graphing the Function

To graph the function, we need to graph each sub-function separately and then combine them.

Graphing the Sub-Function for $x < 0$

The sub-function $f(x) = \frac{1}{4}x$ is a linear function with a slope of $\frac{1}{4}$ and a y-intercept of $0$. To graph this function, we can use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

y = (1/4)x

We can graph this function by plotting two points on the coordinate plane: $(0, 0)$ and $(-4, -1)$. We can then draw a line through these two points to represent the graph of the function.

Graphing the Sub-Function for $x \geq 0$

The sub-function $f(x) = x + 2$ is also a linear function, but with a slope of $1$ and a y-intercept of $2$. To graph this function, we can use the slope-intercept form of a linear equation again.

y = x + 2

We can graph this function by plotting two points on the coordinate plane: $(0, 2)$ and $(1, 3)$. We can then draw a line through these two points to represent the graph of the function.

Combining the Sub-Functions

Now that we have graphed each sub-function separately, we can combine them to get the final graph of the piecewise function. We can do this by drawing a vertical line at $x = 0$ to separate the two sub-functions.

The Final Graph

The final graph of the piecewise function is a combination of the two sub-functions, with a vertical line at $x = 0$ to separate them.

+---------------+
|               |
|  x < 0       |
|  y = (1/4)x  |
|               |
+---------------+
|               |
|  x >= 0      |
|  y = x + 2   |
|               |
+---------------+

Conclusion

Graphing a piecewise function requires understanding the behavior of each sub-function on different intervals of the domain. By graphing each sub-function separately and then combining them, we can get the final graph of the piecewise function. In this article, we used the given function $f(x) = \begin{cases} \frac{1}{4}x, & \text{for } x \ \textless \ 0 \\ x + 2, & \text{for } x \geq 0 \end{cases}$ as an example to illustrate this process.

Choosing the Correct Graph

Now that we have graphed the piecewise function, we can choose the correct graph from the options provided.

A.

B.

C.

D.

The correct graph is the one that matches the final graph we obtained in the previous section.

Discussion

Graphing a piecewise function can be a challenging task, especially when dealing with multiple sub-functions. However, by breaking down the problem into smaller steps and understanding the behavior of each sub-function, we can get the final graph of the piecewise function.

In this article, we used the given function $f(x) = \begin{cases} \frac{1}{4}x, & \text{for } x \ \textless \ 0 \\ x + 2, & \text{for } x \geq 0 \end{cases}$ as an example to illustrate the process of graphing a piecewise function. We hope that this article has provided a clear and concise explanation of this process, and that it has helped you to understand how to graph a piecewise function.

References

• [1] "Graphing Piecewise Functions" by Math Open Reference
• [2] "Piecewise Functions" by Khan Academy
• [3] "Graphing Piecewise Functions" by Purplemath

Additional Resources

• [1] "Graphing Piecewise Functions" by Mathway
• [2] "Piecewise Functions" by Wolfram Alpha
• [3] "Graphing Piecewise Functions" by IXL
  Q&A: Graphing Piecewise Functions =====================================

Frequently Asked Questions

Q: What is a piecewise function?

A: A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q: How do I graph a piecewise function?

A: To graph a piecewise function, you need to graph each sub-function separately and then combine them. You can do this by drawing a vertical line at the point where the sub-functions change.

Q: What if I have multiple sub-functions?

A: If you have multiple sub-functions, you can graph each one separately and then combine them. Make sure to draw a vertical line at the point where each sub-function changes.

Q: How do I choose the correct graph?

A: To choose the correct graph, you need to look at the options provided and see which one matches the final graph you obtained.

Q: What if I'm not sure which graph is correct?

A: If you're not sure which graph is correct, you can try graphing the function again or looking at the options provided to see if you can find a match.

Q: Can I use a graphing calculator to graph a piecewise function?

A: Yes, you can use a graphing calculator to graph a piecewise function. However, make sure to enter the function correctly and adjust the settings as needed.

Q: How do I enter a piecewise function into a graphing calculator?

A: To enter a piecewise function into a graphing calculator, you need to use the "piecewise" or "if-then" function. The exact steps may vary depending on the calculator you're using.

Q: Can I graph a piecewise function on a computer?

A: Yes, you can graph a piecewise function on a computer using a graphing software or a online graphing tool.

Q: How do I graph a piecewise function on a computer?

A: To graph a piecewise function on a computer, you need to use a graphing software or online graphing tool. The exact steps may vary depending on the software or tool you're using.

Q: What if I'm having trouble graphing a piecewise function?

A: If you're having trouble graphing a piecewise function, you can try breaking down the problem into smaller steps, looking at the options provided, or seeking help from a teacher or tutor.

Q: Can I graph a piecewise function with multiple variables?

A: Yes, you can graph a piecewise function with multiple variables. However, the process may be more complex and require additional steps.

Q: How do I graph a piecewise function with multiple variables?

A: To graph a piecewise function with multiple variables, you need to use a graphing software or online graphing tool that can handle multiple variables. The exact steps may vary depending on the software or tool you're using.

Conclusion

Graphing a piecewise function can be a challenging task, but with practice and patience, you can master it. Remember to break down the problem into smaller steps, look at the options provided, and seek help when needed. Good luck!

References

• [1] "Graphing Piecewise Functions" by Math Open Reference
• [2] "Piecewise Functions" by Khan Academy
• [3] "Graphing Piecewise Functions" by Purplemath

Additional Resources

• [1] "Graphing Piecewise Functions" by Mathway
• [2] "Piecewise Functions" by Wolfram Alpha
• [3] "Graphing Piecewise Functions" by IXL