Sure, let's graph the function [tex]\( f(x) = \frac{3}{4}x - 2 \)[/tex] step-by-step.
1. Identify the form of the function:
The function [tex]\( f(x) = \frac{3}{4}x - 2 \)[/tex] is a linear function and can be written in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Determine the slope and y-intercept:
- The slope [tex]\( m \)[/tex] is [tex]\( \frac{3}{4} \)[/tex]. This means for every 4 units you move horizontally to the right, you move 3 units vertically up.
- The y-intercept [tex]\( b \)[/tex] is -2. This is the point where the line crosses the y-axis.
3. Plot the y-intercept:
Start by plotting the point (0, -2) on the graph. This is where the line will intersect the y-axis.
4. Use the slope to find another point:
From the y-intercept (0, -2), use the slope [tex]\( \frac{3}{4} \)[/tex]:
- Move 4 units to the right (positive x direction).
- Move 3 units up (positive y direction).
This gives you a second point at (4, 1).
5. Plot the second point:
Plot the point (4, 1) on the graph.
6. Draw the line:
Draw a straight line through the points (0, -2) and (4, 1). Extend the line in both directions to cover the full range of the graph.
Here is the step-by-step graphical representation of the function [tex]\( f(x) = \frac{3}{4}x - 2 \)[/tex]:
[tex]\[
\begin{aligned}
&\bullet \text{ Plot (0, -2)} \text{ (y-intercept)} \\
&\bullet \text{ Plot another point using the slope (4, 1)} \\
&\bullet \text{ Draw the line passing through both points}
\end{aligned}
\][/tex]
This is how the graph of [tex]\( f(x) = \frac{3}{4}x - 2 \)[/tex] looks like:
1. Plot the Point (0, -2):
- On the graph, mark the point where [tex]\( x = 0 \)[/tex] and [tex]\( y = -2 \)[/tex].
2. Plot the Point (4, 1):
- From the point (0, -2), move horizontally 4 units to the right, and vertically 3 units up. This gets you to the point (4, 1).
- Mark this point on the graph.
3. Draw the Line:
- Connect these two points with a straight line extending it in both directions.
The resulting graph should show a line passing through the points (0, -2) and (4, 1), which accurately represents the linear function [tex]\( f(x) = \frac{3}{4}x - 2 \)[/tex].
