Answer :
To graph the exponential function [tex]\( f(x) \)[/tex] using the given table values, follow these steps:
1. Understand the Data:
The table provides discrete points on the function. Here are the pairs of [tex]\((x, y)\)[/tex] coordinates:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & -7.75 \\ \hline -2 & -7.5 \\ \hline -1 & -7 \\ \hline 0 & -6 \\ \hline 1 & -4 \\ \hline 2 & 0 \\ \hline 3 & 8 \\ \hline \end{array} \][/tex]
2. Setup Your Coordinate Plane:
Create a graph with [tex]\( x \)[/tex]-axis (horizontal) and [tex]\( y \)[/tex]-axis (vertical). You will need to cover the range of [tex]\( x \)[/tex] from -3 to 3 and [tex]\( y \)[/tex] from -8 to 8, based on the provided point values.
3. Plot the Points:
Using the coordinate pairs, plot each point on the graph:
- Plot [tex]\((-3, -7.75)\)[/tex]
- Plot [tex]\((-2, -7.5)\)[/tex]
- Plot [tex]\((-1, -7)\)[/tex]
- Plot [tex]\((0, -6)\)[/tex]
- Plot [tex]\((1, -4)\)[/tex]
- Plot [tex]\((2, 0)\)[/tex]
- Plot [tex]\((3, 8)\)[/tex]
4. Draw the Curve:
After plotting the points, draw a smooth curve that passes through all the points. Since these points represent an exponential function, the curve should show the typical shape of an exponential curve which increases rapidly after passing through the point where [tex]\( y = 0 \)[/tex].
5. Analyze the Graph:
As you draw the curve, observe that:
- For [tex]\( x < 0 \)[/tex], the values of [tex]\( y \)[/tex] are negative and decreasing slowly.
- At [tex]\( x = 0 \)[/tex], [tex]\( y \)[/tex] is -6.
- For [tex]\( x > 0 \)[/tex], the values of [tex]\( y \)[/tex] increase rapidly, especially from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex].
6. Final Touches:
Ensure your curve is smooth and accurately represents the shape suggested by the data points. Label the axes, and provide a title such as "Graph of the Exponential Function".
Here is a rough sketch of the plot:
```
10|
9|
8|
7|
6|
5|
4|
3|
2|
1|
0|
-1|
-2|
-3|
-4|
-5|
-6|
-7|
-8|
-9|________________________________________________*__________
-3 -2 -1 0 1 2 3
```
By following these steps, you can accurately graph the exponential function using the provided table values.
1. Understand the Data:
The table provides discrete points on the function. Here are the pairs of [tex]\((x, y)\)[/tex] coordinates:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & -7.75 \\ \hline -2 & -7.5 \\ \hline -1 & -7 \\ \hline 0 & -6 \\ \hline 1 & -4 \\ \hline 2 & 0 \\ \hline 3 & 8 \\ \hline \end{array} \][/tex]
2. Setup Your Coordinate Plane:
Create a graph with [tex]\( x \)[/tex]-axis (horizontal) and [tex]\( y \)[/tex]-axis (vertical). You will need to cover the range of [tex]\( x \)[/tex] from -3 to 3 and [tex]\( y \)[/tex] from -8 to 8, based on the provided point values.
3. Plot the Points:
Using the coordinate pairs, plot each point on the graph:
- Plot [tex]\((-3, -7.75)\)[/tex]
- Plot [tex]\((-2, -7.5)\)[/tex]
- Plot [tex]\((-1, -7)\)[/tex]
- Plot [tex]\((0, -6)\)[/tex]
- Plot [tex]\((1, -4)\)[/tex]
- Plot [tex]\((2, 0)\)[/tex]
- Plot [tex]\((3, 8)\)[/tex]
4. Draw the Curve:
After plotting the points, draw a smooth curve that passes through all the points. Since these points represent an exponential function, the curve should show the typical shape of an exponential curve which increases rapidly after passing through the point where [tex]\( y = 0 \)[/tex].
5. Analyze the Graph:
As you draw the curve, observe that:
- For [tex]\( x < 0 \)[/tex], the values of [tex]\( y \)[/tex] are negative and decreasing slowly.
- At [tex]\( x = 0 \)[/tex], [tex]\( y \)[/tex] is -6.
- For [tex]\( x > 0 \)[/tex], the values of [tex]\( y \)[/tex] increase rapidly, especially from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex].
6. Final Touches:
Ensure your curve is smooth and accurately represents the shape suggested by the data points. Label the axes, and provide a title such as "Graph of the Exponential Function".
Here is a rough sketch of the plot:
```
10|
9|
8|
7|
6|
5|
4|
3|
2|
1|
0|
-1|
-2|
-3|
-4|
-5|
-6|
-7|
-8|
-9|________________________________________________*__________
-3 -2 -1 0 1 2 3
```
By following these steps, you can accurately graph the exponential function using the provided table values.
