To graph the exponential function [tex]\( k(x) \)[/tex] based on the provided table, follow these steps:
1. Identify the Points:
The table provides pairs [tex]\((x, y)\)[/tex] representing points on the graph. These pairs are:
[tex]\[
(-3, -5.96875), (-2, -5.875), (-1, -5.5), (0, -4), (1, 2), (2, 26), (3, 122)
\][/tex]
2. Set Up Axes:
- Draw a horizontal axis (x-axis) and label it.
- Draw a vertical axis (y-axis) and label it.
- Choose appropriate scales for both axes to accommodate all the x and y values from the table. Ensure the range of x-values (from -3 to 3) and y-values (from approximately -6 to 122) are properly represented.
3. Plot the Points:
Plot each of the points on the coordinate plane. Mark each point clearly.
- [tex]\((-3, -5.96875)\)[/tex]
- [tex]\((-2, -5.875)\)[/tex]
- [tex]\((-1, -5.5)\)[/tex]
- [tex]\((0, -4)\)[/tex]
- [tex]\((1, 2)\)[/tex]
- [tex]\((2, 26)\)[/tex]
- [tex]\((3, 122)\)[/tex]
4. Connect the Points:
Once all the points are plotted, draw a smooth curve through the points to reflect the behavior of the exponential function. The curve should demonstrate the rapid increase in y-values as x increases, which is characteristic of exponential functions.
5. Label the Graph:
- Add a title to the graph such as "Graph of the Exponential Function [tex]\( k(x) \)[/tex]".
- Label the x-axis as [tex]\( x \)[/tex].
- Label the y-axis as [tex]\( y \)[/tex].
Here is a conceptual sketch of the graph based on the provided points:
[tex]\[
\begin{array}{cc}
x & y \\
-3 & -5.96875 \\
-2 & -5.875 \\
-1 & -5.5 \\
0 & -4 \\
1 & 2 \\
2 & 26 \\
3 & 122 \\
\end{array}
\][/tex]
The y-values rapidly increase as the x-values move from positive to negative, reflecting the exponential nature of the function. The points form a pattern that rises very steeply as [tex]\( x \)[/tex] increases, which you should replicate with a smooth curve on your graph.
Ensure your final graph is neat, with clearly marked and scaled axes to illustrate the exponential growth reflected by these points.
