Graph This Function:$\[ F(x)=\left\{\begin{array}{ll} -x+3 & \text{if } X \leq 6 \\ 3x-21 & \text{if } X\ \textgreater \ 6 \end{array}\right. \\]Select Points On The Graph To Plot Them. Select Point Fill To Change A Point From Closed To

Introduction

Piecewise functions are a type of 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 function $f(x)=\left\{\begin{array}{ll} -x+3 & \text{if } x \leq 6 \\ 3x-21 & \text{if } x\ \textgreater \ 6 \end{array}\right.$ as an example.

Understanding Piecewise Functions

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

$$f(x)=\left\{\begin{array}{ll} f_1(x) & \text{if } x \in D_1 \\ f_2(x) & \text{if } x \in D_2 \\ \vdots & \vdots \end{array}\right.$$

where $f_1(x)$, $f_2(x)$, etc. are the sub-functions, and $D_1$, $D_2$, etc. are the intervals of the domain where each sub-function is applied.

Graphing the Piecewise Function

To graph the piecewise function $f(x)=\left\{\begin{array}{ll} -x+3 & \text{if } x \leq 6 \\ 3x-21 & \text{if } x\ \textgreater \ 6 \end{array}\right.$, we need to graph each sub-function separately and then combine them.

Graphing the First Sub-Function

The first sub-function is $f_1(x)=-x+3$, which is a linear function. To graph this function, we can use the slope-intercept form of a linear function, which is $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept.

The slope of the first sub-function is $-1$, and the y-intercept is $3$. Therefore, the equation of the first sub-function is:

$$y=-x+3$$

To graph this function, we can plot two points on the graph: one point at $x=0$ and $y=3$, and another point at $x=6$ and $y=-3$.

Graphing the Second Sub-Function

The second sub-function is $f_2(x)=3x-21$, which is also a linear function. To graph this function, we can use the slope-intercept form of a linear function, which is $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept.

The slope of the second sub-function is $3$, and the y-intercept is $-21$. Therefore, the equation of the second sub-function is:

$$y=3x-21$$

To graph this function, we can plot two points on the graph: one point at $x=6$ and $y=9$, and another point at $x=0$ and $y=-21$.

Combining the Sub-Functions

Now that we have graphed each sub-function separately, we can combine them to get the graph of the piecewise function.

The graph of the piecewise function is a combination of the two sub-functions, with the first sub-function applied to the interval $x \leq 6$ and the second sub-function applied to the interval $x > 6$.

Selecting Points to Plot

To plot the graph of the piecewise function, we need to select points on the graph to plot them. We can select points on the graph by using the following steps:

1. Identify the intervals of the domain where each sub-function is applied.
2. Plot two points on the graph for each sub-function, one point at the beginning of the interval and another point at the end of the interval.
3. Use the points to draw the graph of the piecewise function.

Selecting "Point Fill"

To change a point from closed to open, we can select "Point fill" from the graphing options. This will change the point from a closed circle to an open circle.

Conclusion

In this article, we have explored how to graph a piecewise function, using the function $f(x)=\left\{\begin{array}{ll} -x+3 & \text{if } x \leq 6 \\ 3x-21 & \text{if } x\ \textgreater \ 6 \end{array}\right.$ as an example. We have graphed each sub-function separately and then combined them to get the graph of the piecewise function. We have also discussed how to select points on the graph to plot them and how to change a point from closed to open using "Point fill".

Graphing Piecewise Functions: Tips and Tricks

Here are some tips and tricks for graphing piecewise functions:

• Make sure to identify the intervals of the domain where each sub-function is applied.
• Plot two points on the graph for each sub-function, one point at the beginning of the interval and another point at the end of the interval.
• Use the points to draw the graph of the piecewise function.
• Select "Point fill" to change a point from closed to open.
• Use the graphing options to customize the graph and make it easier to read.

Common Mistakes to Avoid

Here are some common mistakes to avoid when graphing piecewise functions:

• Failing to identify the intervals of the domain where each sub-function is applied.
• Plotting points on the graph that are not on the function.
• Failing to use the points to draw the graph of the piecewise function.
• Not selecting "Point fill" to change a point from closed to open.
• Not using the graphing options to customize the graph and make it easier to read.

Conclusion

Q: What is a piecewise function?

A: A piecewise function is a type of 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 use the slope-intercept form of a linear function to graph each sub-function.

Q: What are the intervals of the domain where each sub-function is applied?

A: The intervals of the domain where each sub-function is applied are the intervals that are defined by the piecewise function. For example, if the piecewise function is defined as:

$$f(x)=\left\{\begin{array}{ll} -x+3 & \text{if } x \leq 6 \\ 3x-21 & \text{if } x\ \textgreater \ 6 \end{array}\right.$$

then the intervals of the domain where each sub-function is applied are $x \leq 6$ and $x > 6$.

Q: How do I plot points on the graph of a piecewise function?

A: To plot points on the graph of a piecewise function, you need to identify the intervals of the domain where each sub-function is applied and plot two points on the graph for each sub-function, one point at the beginning of the interval and another point at the end of the interval.

Q: What is the difference between a closed and open point on a graph?

A: A closed point on a graph is a point that is marked with a closed circle, while an open point on a graph is a point that is marked with an open circle. You can change a point from closed to open by selecting "Point fill" from the graphing options.

Q: How do I customize the graph of a piecewise function?

A: You can customize the graph of a piecewise function by using the graphing options to change the appearance of the graph. For example, you can change the color of the graph, add labels to the axes, and add a title to the graph.

Q: What are some common mistakes to avoid when graphing piecewise functions?

A: Some common mistakes to avoid when graphing piecewise functions include:

• Failing to identify the intervals of the domain where each sub-function is applied
• Plotting points on the graph that are not on the function
• Failing to use the points to draw the graph of the piecewise function
• Not selecting "Point fill" to change a point from closed to open
• Not using the graphing options to customize the graph and make it easier to read

Q: How can I practice graphing piecewise functions?

A: You can practice graphing piecewise functions by working through examples and exercises in a textbook or online resource. You can also try graphing piecewise functions on your own using a graphing calculator or online graphing tool.

Q: What are some real-world applications of piecewise functions?

A: Piecewise functions have many real-world applications, including:

• Modeling population growth and decline
• Modeling the cost of goods and services
• Modeling the behavior of physical systems, such as the motion of an object under the influence of gravity
• Modeling the behavior of financial systems, such as the stock market

Conclusion

Graphing piecewise functions can be a challenging task, but with practice and patience, you can become proficient in graphing these types of functions. Remember to identify the intervals of the domain where each sub-function is applied, plot two points on the graph for each sub-function, and use the points to draw the graph of the piecewise function. By following these steps and using the graphing options to customize the graph, you can create a clear and accurate graph of the piecewise function.