To graph the function [tex]\( g(x) = -8x \)[/tex] and its asymptote, follow these steps:
### Step-by-Step Solution
1. Understanding the Function:
- The function [tex]\( g(x) = -8x \)[/tex] is a linear function.
- Its graph will be a straight line with a slope of [tex]\(-8\)[/tex]. This means, for every 1 unit you move to the right along the x-axis, you move 8 units down along the y-axis.
2. Identifying the Asymptote:
- Linear functions do not have vertical or horizontal asymptotes in the traditional sense. However, we can consider the x-axis (y=0) as an "asymptote" to the line as it gets very close to the x-axis for large values of |x|.
- For the purposes of this graph, we can draw the x-axis as a dashed line to indicate this "asymptote."
3. Plot Points from the Function:
- Choose a set of x-values to compute corresponding y-values using the function [tex]\( g(x) = -8x \)[/tex].
Points to plot could include:
- [tex]\( g(0) = -8 \cdot 0 = 0 \)[/tex]
- [tex]\( g(1) = -8 \cdot 1 = -8 \)[/tex]
- [tex]\( g(-1) = -8 \cdot (-1) = 8 \)[/tex]
- [tex]\( g(2) = -8 \cdot 2 = -16 \)[/tex]
- [tex]\( g(-2) = -8 \cdot (-2) = 16 \)[/tex]
- Plot these points on the graph.
4. Drawing the Graph:
- Draw the straight line through the points plotted above.
5. Indicating the Asymptote:
- Draw a dashed line along the x-axis, indicating [tex]\( y = 0 \)[/tex].
### Graph Representation
Here's the detailed representation of how it would look:
#### Graph:
- The x-axis ranges from [tex]\(-10\)[/tex] to [tex]\(10\)[/tex] for a broad view.
- The y-axis ranges appropriately to capture the plotted points, say from [tex]\(-80\)[/tex] to [tex]\(80\)[/tex], given the slope of [tex]\(-8\)[/tex].
| x | g(x) |
|------|---------|
| -2 | 16 |
| -1 | 8 |
| 0 | 0 |
| 1 | -8 |
| 2 | -16 |
### Tool: Graphing the Function
If you are using graphing tools (like Desmos, GeoGebra, or any graphing calculator):
1. Input the Function:
- Enter the equation [tex]\( g(x) = -8x \)[/tex].
2. Plot the Points:
- Enable plotting of specific points if your tool supports it.
3. Draw the Asymptote:
- Draw or plot the dashed line along the x-axis ( [tex]\( y = 0 \)[/tex] ).
4. Final Representation:
- Make sure the graph scale is even, and labels are clear for both axes.
- Include gridlines if supported for better readability.
### Visual Example
Example: Desmos graph might have lines and points clearly plotted. Here's what you would expect:
1. Straight line descending from left to right.
2. Dashed line along the x-axis.
This method will help visualize and understand how [tex]\( g(x) = -8x \)[/tex] behaves, how steep the slope is, and how close it gets to the x-axis without actually intersecting it at large values of [tex]\(|x|\)[/tex].
