To graph the line that represents a proportional relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] with a unit rate of change of [tex]\( \frac{3}{8} \)[/tex], follow these steps:
1. Understand the Unit Rate (Slope):
- The unit rate of change [tex]\( \frac{3}{8} \)[/tex] means that for every increase of 1 unit in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by [tex]\( \frac{3}{8} \)[/tex] units.
- This unit rate of change is also known as the slope of the line.
2. Write the Equation:
- In a proportional relationship, the equation of the line is typically written in the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the unit rate or slope.
- Here, the slope [tex]\( k = \frac{3}{8} \)[/tex].
- So, the equation of the line is:
[tex]\[
\boxed{y = \frac{3}{8}x}
\][/tex]
3. Plotting the Line:
- Identify key points to plot on the graph. You can use the equation [tex]\( y = \frac{3}{8}x \)[/tex] to calculate these points.
- For example:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = \frac{3}{8} \cdot 0 = 0
\][/tex]
So, the point [tex]\((0,0)\)[/tex] is on the line.
- When [tex]\( x = 8 \)[/tex]:
[tex]\[
y = \frac{3}{8} \cdot 8 = 3
\][/tex]
So, the point [tex]\((8, 3)\)[/tex] is on the line.
- When [tex]\( x = 4 \)[/tex]:
[tex]\[
y = \frac{3}{8} \cdot 4 = 1.5
\][/tex]
So, the point [tex]\((4, 1.5)\)[/tex] is on the line.
4. Draw the Line:
- Plot the points [tex]\((0,0)\)[/tex], [tex]\((8,3)\)[/tex], and [tex]\((4,1.5)\)[/tex] on a coordinate plane.
- Draw a straight line through these points, extending it in both directions to cover the graph area.
Your graph should look like this:
```
y
|
| . (8, 3)
| /
| /
| .(4, 1.5)
| /
| .(0, 0)
-------------------------------- x
```
This straight line represents the proportional relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] with a unit rate of change of [tex]\( \frac{3}{8} \)[/tex].
In conclusion, the equation of the line is [tex]\( \boxed{y = \frac{3}{8}x} \)[/tex].
