Question

Graph the line that represents a proportional relationship between

and

where the unit rate of change of

with respect to

is


. In other words, a change of

unit in

corresponds to a change of


units in

.
Graph the line and write the equation of the line.

Answer 1

To graph a line that represents a proportional relationship between two variables, we'll first need to establish what the variables are and the unit rate of change (the slope).

Let's assume:
- The variable \( x \) represents one quantity.
- The variable \( y \) represents another quantity.
- The unit rate of change of \( y \) with respect to \( x \) is represented as \( k \), where a change of 1 unit in \( x \) corresponds to a change of \( k \) units in \( y \).

### Step 1: Define the Relationship
The relationship can be defined as:
\[
y = kx
\]
This equation states that \( y \) is directly proportional to \( x \), with \( k \) being the constant of proportionality (the slope of the line).

### Step 2: Choose a Value for \( k \)
For the sake of this example, let's choose \( k = 2 \). This means:
\[
y = 2x
\]

### Step 3: Create a Table of Values
Let's create a table for \( x \) and \( y \):

| \( x \) | \( y \) |
|---------|---------|
| 0 | 0 |
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |

### Step 4: Plot the Points
Now, we can plot the points from the table:
1. \( (0, 0) \)
2. \( (1, 2) \)
3. \( (2, 4) \)
4. \( (3, 6) \)
5. \( (4, 8) \)

### Step 5: Draw the Line
Draw a straight line through all the points plotted on the graph.

### Graph Representation
The line starts at the origin (0, 0) and goes through (1, 2), (2, 4), (3, 6), and (4, 8), extending infinitely in both directions, maintaining the slope of 2.

### Step 6: Write the Equation of the Line
The equation of the line based on our chosen unit rate of change is:
\[
y = 2x
\]

### Conclusion
You have now graphed the line representing a proportional relationship between \( x \) and \( y \), with the equation \( y = 2x \). If you require a different unit rate of change, you can replace \( k \) with your desired value and follow the same steps.