Certainly! Let's graph the function [tex]\( f(x) = 3x \)[/tex] step-by-step.
### Step 1: Understanding the Function
The function [tex]\( f(x) = 3x \)[/tex] is a linear function, which means its graph will be a straight line. The general form of a linear function is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
For [tex]\( f(x) = 3x \)[/tex]:
- The slope ([tex]\( m \)[/tex]) is 3. This means that for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 3 units.
- The y-intercept ([tex]\( b \)[/tex]) is 0. This means that the line passes through the origin (0, 0).
### Step 2: Plotting Points
To graph the function, we can plot a few points by substituting [tex]\( x \)[/tex]-values into the function to find the corresponding [tex]\( y \)[/tex]-values.
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 3 \times 0 = 0
\][/tex]
So, the point is (0, 0).
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3 \times 1 = 3
\][/tex]
So, the point is (1, 3).
- When [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 3 \times -1 = -3
\][/tex]
So, the point is (-1, -3).
### Step 3: Drawing the Line
Now we can use these points to draw the line:
1. Plot the points (0, 0), (1, 3), and (-1, -3) on the Cartesian plane.
2. Draw a straight line through these points extending in both directions.
### Step 4: Labeling the Graph
- Label the x-axis and y-axis.
- Mark and label the points you used to plot the line.
- Label the line with the function [tex]\( f(x) = 3x \)[/tex].
### Example Plot
[tex]\[
\begin{array}{r|r}
x & f(x) \\
\hline
-2 & -6 \\
-1 & -3 \\
0 & 0 \\
1 & 3 \\
2 & 6 \\
\end{array}
\][/tex]
Plotting these points on a graph, your linear function [tex]\( f(x) = 3x \)[/tex] will look like this:
[tex]\[
\begin{array}{c}
| \\
6 | \\
5 | \\
4 | \\
3 | \cdot (1, 3) \\
2 | \\
1 | \\
0 | \cdot (0, 0) \\
-1 | \\
-2 | \\
-3 | \cdot (-1, -3) \\
-4 | \\
-5 | \\
-6 | \\
\end{array}
\][/tex]
The line passes through these plotted points. The slope [tex]\( m = 3 \)[/tex] makes the line steep, indicating that [tex]\( y \)[/tex] increases three times as fast as [tex]\( x \)[/tex].
### Conclusion
The function [tex]\( f(x) = 3x \)[/tex] is represented graphically as a straight line passing through the origin with a slope of 3.
