First, let's break down the problem step-by-step.
### Part A: Drawing the Graph
#### Step 1: Understand the Data
We have the following data regarding the distance Ami rode her bike for different amounts of time.
| Time (minutes) | 5 | 10 | 15 | 20 | 25 | 30 |
|----------------|-----|-----|-----|-----|-----|-----|
| Distance (miles)| 0.75| 1.5 | 2.25| 3 | 3.75| 4.5 |
#### Step 2: Set Up the Axes
- The [tex]\(x\)[/tex]-axis will represent the time in minutes.
- The [tex]\(y\)[/tex]-axis will represent the distance in miles.
- Make sure to label the axes and provide an appropriate scale.
#### Step 3: Plot the Points
Plot each pair of [tex]\((x, y)\)[/tex] coordinates from the table:
[tex]\[
(5, 0.75), (10, 1.5), (15, 2.25), (20, 3), (25, 3.75), (30, 4.5)
\][/tex]
### Graph Description
To draw the graph:
1. Draw a horizontal line and label it as the [tex]\(x\)[/tex]-axis (Time in minutes).
2. Draw a vertical line and label it as the [tex]\(y\)[/tex]-axis (Distance in miles).
3. Mark points on the [tex]\(x\)[/tex]-axis at intervals of 5 (5, 10, 15, 20, 25, 30).
4. Mark points on the [tex]\(y\)[/tex]-axis at intervals of 0.75 (0.75, 1.5, 2.25, 3, 3.75, 4.5).
5. Plot the points [tex]\((5, 0.75)\)[/tex], [tex]\((10, 1.5)\)[/tex], [tex]\((15, 2.25)\)[/tex], [tex]\((20, 3)\)[/tex], [tex]\((25, 3.75)\)[/tex], and [tex]\((30, 4.5)\)[/tex].
6. Connect the points with a straight line.
Here's a rough sketch:
```
Distance (miles)
|
4.5 |
4.0 |
3.75 |
3.0 |
2.25 |
1.5 |
0.75 |
0|__________________________________________________________
5 10 15 20 25 30
Time (minutes)
```
### Checking for Proportional Relationship
A proportional relationship means that the ratio between the time and distance should remain constant. Let's calculate the ratio of distance to time for each pair of data points:
[tex]\[
\frac{0.75}{5} = 0.15, \quad \frac{1.5}{10} = 0.15, \quad \frac{2.25}{15} = 0.15, \quad \frac{3}{20} = 0.15, \quad \frac{3.75}{25} = 0.15, \quad \frac{4.5}{30} = 0.15
\][/tex]
The ratio of [tex]\(\frac{\text{distance}}{\text{time}} = 0.15\)[/tex] is the same for all pairs of points.
### Conclusion
Since the ratio of distance to time is constant, the graph shows a straight line passing through the origin (0,0). This confirms that the graph does show a proportional relationship between the time and distance Ami rode her bike. The graph is a straight line indicating that as time increases, the distance increases proportionally.
