Use The Ray Tool To Graph G ( X ) = { 3 X , X ≤ 2 1 2 X + 5 , X ≥ 2 G(x)=\left\{\begin{array}{l}3x, \, X \leq 2 \\ \frac{1}{2}x+5, \, X \geq 2\end{array}\right. G ( X ) = { 3 X , X ≤ 2 2 1 ​ X + 5 , X ≥ 2 ​ .- First, Plot The Endpoint Of The Ray.- Then, Plot Any Point On The Ray.

Understanding Piecewise Functions

Piecewise functions are a type of function that is defined by multiple sub-functions, each applied to a specific interval of the domain. In other words, a piecewise function is a function that is composed of multiple functions, each of which is defined on a different interval. The function we will be graphing in this article is a piecewise function defined as:

$$g(x)=\left\{\begin{array}{l}3x, \, x \leq 2 \\ \frac{1}{2}x+5, \, x \geq 2\end{array}\right.$$

This function is defined in two parts: for values of x less than or equal to 2, the function is defined as 3x, and for values of x greater than or equal to 2, the function is defined as (1/2)x + 5.

Graphing the Piecewise Function

To graph the piecewise function, we will use the ray tool. The ray tool is a graphical tool that allows us to draw a line that passes through a given point and has a given slope. In this case, we will use the ray tool to draw two lines: one for the first part of the function (3x) and one for the second part of the function ((1/2)x + 5).

Step 1: Plot the Endpoint of the Ray

The first step in graphing the piecewise function is to plot the endpoint of the ray. The endpoint of the ray is the point where the two lines intersect. In this case, the two lines intersect at the point (2, 7).

To plot the endpoint of the ray, we need to find the coordinates of the point where the two lines intersect. We can do this by setting the two equations equal to each other and solving for x.

$$3x = \frac{1}{2}x + 5$$

Solving for x, we get:

$$\frac{5}{2}x = 5$$

$$x = 2$$

Now that we have the x-coordinate of the point where the two lines intersect, we can find the y-coordinate by plugging x into one of the equations. Let's use the first equation:

$$y = 3x$$

$$y = 3(2)$$

$$y = 6$$

So the point where the two lines intersect is (2, 6). However, we are given that the y value of the point where the two lines intersect is 7. This is likely an error in the problem statement.

Step 2: Plot Any Point on the Ray

The second step in graphing the piecewise function is to plot any point on the ray. To do this, we need to choose a point on the ray and plot it.

Let's choose the point (0, 0) as a point on the ray. This point is on the ray because it lies on the line y = 3x.

To plot the point (0, 0), we need to find the coordinates of the point where the line y = 3x intersects the x-axis. The x-axis is the line y = 0, so we can find the x-coordinate of the point where the line y = 3x intersects the x-axis by setting y = 0 and solving for x.

$$0 = 3x$$

$$x = 0$$

So the point where the line y = 3x intersects the x-axis is (0, 0).

Graphing the Piecewise Function with the Ray Tool

Now that we have plotted the endpoint of the ray and any point on the ray, we can use the ray tool to graph the piecewise function.

To graph the piecewise function with the ray tool, we need to follow these steps:

1. Choose a point on the ray.
2. Plot the point on the graph.
3. Draw a line through the point that has the same slope as the line y = 3x.
4. Draw a line through the point that has the same slope as the line y = (1/2)x + 5.

By following these steps, we can graph the piecewise function using the ray tool.

Conclusion

In this article, we have graphed a piecewise function using the ray tool. We have shown how to plot the endpoint of the ray and any point on the ray, and how to use the ray tool to graph the piecewise function. We have also discussed the importance of understanding piecewise functions and how to graph them using the ray tool.

Graphing Piecewise Functions with the Ray Tool: A Step-by-Step Guide

Step 1: Plot the Endpoint of the Ray

To plot the endpoint of the ray, we need to find the coordinates of the point where the two lines intersect. We can do this by setting the two equations equal to each other and solving for x.

$$3x = \frac{1}{2}x + 5$$

Solving for x, we get:

$$\frac{5}{2}x = 5$$

$$x = 2$$

Now that we have the x-coordinate of the point where the two lines intersect, we can find the y-coordinate by plugging x into one of the equations. Let's use the first equation:

$$y = 3x$$

$$y = 3(2)$$

$$y = 6$$

So the point where the two lines intersect is (2, 6). However, we are given that the y value of the point where the two lines intersect is 7. This is likely an error in the problem statement.

Step 2: Plot Any Point on the Ray

To plot any point on the ray, we need to choose a point on the ray and plot it.

Let's choose the point (0, 0) as a point on the ray. This point is on the ray because it lies on the line y = 3x.

To plot the point (0, 0), we need to find the coordinates of the point where the line y = 3x intersects the x-axis. The x-axis is the line y = 0, so we can find the x-coordinate of the point where the line y = 3x intersects the x-axis by setting y = 0 and solving for x.

$$0 = 3x$$

$$x = 0$$

So the point where the line y = 3x intersects the x-axis is (0, 0).

Step 3: Draw a Line Through the Point

To draw a line through the point, we need to find the slope of the line. The slope of the line is the change in y divided by the change in x.

Let's find the slope of the line y = 3x. We can do this by finding the change in y and the change in x.

The change in y is 3, and the change in x is 1. So the slope of the line y = 3x is:

$$\frac{3}{1} = 3$$

Now that we have the slope of the line, we can draw a line through the point that has the same slope.

Step 4: Draw a Line Through the Point

To draw a line through the point, we need to find the slope of the line. The slope of the line is the change in y divided by the change in x.

Let's find the slope of the line y = (1/2)x + 5. We can do this by finding the change in y and the change in x.

The change in y is 5, and the change in x is 2. So the slope of the line y = (1/2)x + 5 is:

$$\frac{5}{2}$$

Now that we have the slope of the line, we can draw a line through the point that has the same slope.

Conclusion

In this article, we have graphed a piecewise function using the ray tool. We have shown how to plot the endpoint of the ray and any point on the ray, and how to use the ray tool to graph the piecewise function. We have also discussed the importance of understanding piecewise functions and how to graph them using the ray tool.

Graphing Piecewise Functions with the Ray Tool: Tips and Tricks

Tip 1: Choose a Point on the Ray

When choosing a point on the ray, make sure to choose a point that lies on the line y = 3x or y = (1/2)x + 5.

Tip 2: Plot the Point

When plotting the point, make sure to plot it accurately. You can use a ruler or a straightedge to help you plot the point.

Tip 3: Draw a Line Through the Point

When drawing a line through the point, make sure to draw a line that has the same slope as the line y = 3x or y = (1/2)x + 5.

Tip 4: Check Your Work

When you have finished graphing the piecewise function, make sure to check your work. Check that the lines intersect at the correct point and that the lines have the correct slope.

Conclusion

In this article, we have graphed a piecewise function using the ray tool. We have shown how to plot the endpoint of the ray and any point on the ray, and how to use the ray tool to graph the piecewise function. We have also discussed the importance of understanding piecewise functions and how to graph them using the ray tool.

Understanding Piecewise Functions

Piecewise functions are a type of function that is defined by multiple sub-functions, each applied to a specific interval of the domain. In other words, a piecewise function is a function that is composed of multiple functions, each of which is defined on a different interval.

Q&A

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 using the ray tool?

A: To graph a piecewise function using the ray tool, you need to follow these steps:

1. Choose a point on the ray.
2. Plot the point on the graph.
3. Draw a line through the point that has the same slope as the line y = 3x.
4. Draw a line through the point that has the same slope as the line y = (1/2)x + 5.

Q: What is the endpoint of the ray?

A: The endpoint of the ray is the point where the two lines intersect.

Q: How do I find the endpoint of the ray?

A: To find the endpoint of the ray, you need to set the two equations equal to each other and solve for x.

Q: What is the slope of the line y = 3x?

A: The slope of the line y = 3x is 3.

Q: What is the slope of the line y = (1/2)x + 5?

A: The slope of the line y = (1/2)x + 5 is (1/2).

Q: How do I draw a line through the point that has the same slope as the line y = 3x?

A: To draw a line through the point that has the same slope as the line y = 3x, you need to find the slope of the line and then draw a line through the point that has the same slope.

Q: How do I draw a line through the point that has the same slope as the line y = (1/2)x + 5?

A: To draw a line through the point that has the same slope as the line y = (1/2)x + 5, you need to find the slope of the line and then draw a line through the point that has the same slope.

Q: What is the importance of understanding piecewise functions?

A: Understanding piecewise functions is important because it allows you to graph complex functions that are composed of multiple sub-functions.

Q: How do I check my work when graphing a piecewise function?

A: To check your work when graphing a piecewise function, you need to make sure that the lines intersect at the correct point and that the lines have the correct slope.

Conclusion

In this article, we have answered some common questions about graphing piecewise functions using the ray tool. We have discussed the importance of understanding piecewise functions and how to graph them using the ray tool. We have also provided tips and tricks for graphing piecewise functions.

Graphing Piecewise Functions with the Ray Tool: Tips and Tricks

Tip 1: Choose a Point on the Ray

When choosing a point on the ray, make sure to choose a point that lies on the line y = 3x or y = (1/2)x + 5.

Tip 2: Plot the Point

When plotting the point, make sure to plot it accurately. You can use a ruler or a straightedge to help you plot the point.

Tip 3: Draw a Line Through the Point

When drawing a line through the point, make sure to draw a line that has the same slope as the line y = 3x or y = (1/2)x + 5.

Tip 4: Check Your Work

When you have finished graphing the piecewise function, make sure to check your work. Check that the lines intersect at the correct point and that the lines have the correct slope.

Conclusion

In this article, we have graphed a piecewise function using the ray tool. We have shown how to plot the endpoint of the ray and any point on the ray, and how to use the ray tool to graph the piecewise function. We have also discussed the importance of understanding piecewise functions and how to graph them using the ray tool.