Given the provided table where the values of [tex]\( x \)[/tex] and [tex]\( y = k(x) \)[/tex] are as follows:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\
\hline
y & 134 & 38 & 14 & 8 & 6.5 & 6.125 & 6.03125 \\
\hline
\end{array}
\][/tex]
we can plot these points on the Cartesian plane to graph the exponential function [tex]\( k(x) \)[/tex].
### Plotting the Points
1. For [tex]\( x = -3 \)[/tex], [tex]\( y = 134 \)[/tex]
2. For [tex]\( x = -2 \)[/tex], [tex]\( y = 38 \)[/tex]
3. For [tex]\( x = -1 \)[/tex], [tex]\( y = 14 \)[/tex]
4. For [tex]\( x = 0 \)[/tex], [tex]\( y = 8 \)[/tex]
5. For [tex]\( x = 1 \)[/tex], [tex]\( y = 6.5 \)[/tex]
6. For [tex]\( x = 2 \)[/tex], [tex]\( y = 6.125 \)[/tex]
7. For [tex]\( x = 3 \)[/tex], [tex]\( y = 6.03125 \)[/tex]
When you graph these points, you'll notice a general trend:
- As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] decreases.
- As [tex]\( x \)[/tex] decreases, [tex]\( y \)[/tex] increases significantly.
### General Shape of the Graph
This set of points suggests that the function [tex]\( k(x) \)[/tex] is an exponential decay function because as the value of [tex]\( x \)[/tex] increases, the value of [tex]\( y \)[/tex] approaches a certain limit (which appears to be a number slightly above 6). The rapid increase in [tex]\( y \)[/tex] as [tex]\( x \)[/tex] becomes more negative is indicative of the nature of exponential growth in the negative direction.
### Graph
Here is a rough sketch of the points and the exponential decay curve associated with them:
```
y-axis
^
|
|
|
|
|
|
|------------------> x-axis
```
In a more precise graph, you would plot the points more accurately. The exponential decay function depicted by the table of values has an asymptote slightly above [tex]\( y = 6 \)[/tex], and it passes through all the provided points.
This graph reflects the decreasing trend as [tex]\( x \)[/tex] increases and the dramatic increase in [tex]\( y \)[/tex] values as [tex]\( x \)[/tex] transitions to more negative values.
