Unit I Worksheet 1: Graphing Practice - Students analyze four data sets to determine mathematical expressions and graph relationships between variables.

Problem Overview:

The task involves analyzing four data sets and determining mathematical expressions that describe the relationships between the variables. The goal is to modify the data so that it plots as a straight line, then use the slope and y-intercept of the linear fit to write appropriate mathematical expressions.

Step-by-Step Solution:

#### Data Set 1:
- Variables: \( V \) (m³) and \( P \) (Pa)
- Data:
| \( V \) (m³) | \( P \) (Pa) |
|-------------|-------------|
| 0.1 | 40 |
| 0.5 | 8 |
| 1 | 4 |
| 2 | 2 |
| 4 | 1 |
| 5 | 0.8 |
| 8 | 0.5 |
| 10 | 0.4 |

Analysis:
The relationship between \( V \) and \( P \) appears to be inversely proportional, i.e., \( P \propto \frac{1}{V} \). To plot this as a straight line, we can take the reciprocal of \( V \) (i.e., \( \frac{1}{V} \)) and plot \( P \) against \( \frac{1}{V} \).

- Modified Data:
| \( \frac{1}{V} \) | \( P \) (Pa) |
|------------------|--------------|
| 10 | 40 |
| 2 | 8 |
| 1 | 4 |
| 0.5 | 2 |
| 0.25 | 1 |
| 0.2 | 0.8 |
| 0.125 | 0.5 |
| 0.1 | 0.4 |

Linear Fit:
Plotting \( P \) vs. \( \frac{1}{V} \) gives a straight line. The slope of this line is the constant of proportionality.

- Slope (\( k \)): From the data, \( P = k \cdot \frac{1}{V} \). For example, when \( \frac{1}{V} = 10 \), \( P = 40 \). Thus, \( k = 40 \).

Mathematical Expression:
\[ P = \frac{40}{V} \]

---

#### Data Set 2:
- Variables: \( t \) (s) and \( x \) (m)
- Data:
| \( t \) (s) | \( x \) (m) |
|------------|-------------|
| 0.1 | 0.03 |
| 2 | 0.12 |
| 5 | 0.75 |
| 1 | 3 |
| 2 | 12 |
| 3 | 27 |
| 4 | 48 |
| 5 | 75 |

Analysis:
The relationship between \( t \) and \( x \) appears quadratic, i.e., \( x \propto t^2 \). To plot this as a straight line, we can plot \( x \) against \( t^2 \).

- Modified Data:
| \( t^2 \) | \( x \) (m) |
|------------|-------------|
| 0.01 | 0.03 |
| 4 | 0.12 |
| 25 | 0.75 |
| 1 | 3 |
| 4 | 12 |
| 9 | 27 |
| 16 | 48 |
| 25 | 75 |

Linear Fit:
Plotting \( x \) vs. \( t^2 \) gives a straight line. The slope of this line is the constant of proportionality.

- Slope (\( k \)): From the data, \( x = k \cdot t^2 \). For example, when \( t^2 = 1 \), \( x = 3 \). Thus, \( k = 3 \).

Mathematical Expression:
\[ x = 3t^2 \]

---

#### Data Set 3:
- Variables: \( A \) (months) and \( W \) (lbs)
- Data:
| \( A \) (months) | \( W \) (lbs) |
|------------------|---------------|
| 1 | 7.3 |
| 2 | 9.4 |
| 3 | 10.5 |
| 4 | 12.0 |
| 5 | 13.0 |
| 6 | 14.3 |
| 7 | 15.2 |
| 8 | 16.7 |

Analysis:
The relationship between \( A \) and \( W \) appears linear. We can directly plot \( W \) against \( A \) and find the slope and y-intercept.

Linear Fit:
Using the first two points:
- When \( A = 1 \), \( W = 7.3 \)
- When \( A = 2 \), \( W = 9.4 \)

Slope (\( m \)):
\[ m = \frac{\Delta W}{\Delta A} = \frac{9.4 - 7.3}{2 - 1} = 2.1 \]

Y-intercept (\( b \)):
Using the point-slope form \( W = mA + b \):
\[ 7.3 = 2.1(1) + b \]
\[ b = 7.3 - 2.1 = 5.2 \]

Mathematical Expression:
\[ W = 2.1A + 5.2 \]

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#### Data Set 4:
- Variables: \( t \) (s) and \( v \) (m/s)
- Data:
| \( t \) (s) | \( v \) (m/s) |
|------------|---------------|
| 0.3 | 10 |
| 1.2 | 20 |
| 2.7 | 30 |
| 4.8 | 40 |
| 7.5 | 50 |
| 10.8 | 60 |
| 14.7 | 70 |
| 19.2 | 80 |

Analysis:
The relationship between \( t \) and \( v \) appears linear. We can directly plot \( v \) against \( t \) and find the slope and y-intercept.

Linear Fit:
Using the first two points:
- When \( t = 0.3 \), \( v = 10 \)
- When \( t = 1.2 \), \( v = 20 \)

Slope (\( m \)):
\[ m = \frac{\Delta v}{\Delta t} = \frac{20 - 10}{1.2 - 0.3} = \frac{10}{0.9} \approx 11.11 \]

Y-intercept (\( b \)):
Using the point-slope form \( v = mt + b \):
\[ 10 = 11.11(0.3) + b \]
\[ 10 = 3.33 + b \]
\[ b = 10 - 3.33 = 6.67 \]

Mathematical Expression:
\[ v = 11.11t + 6.67 \]

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Final Answers:

1. Mathematical Expression #1: \( P = \frac{40}{V} \)
2. Mathematical Expression #2: \( x = 3t^2 \)
3. Mathematical Expression #3: \( W = 2.1A + 5.2 \)
4. Mathematical Expression #4: \( v = 11.11t + 6.67 \)

\[
\boxed{
\begin{aligned}
1. & \quad P = \frac{40}{V} \\
2. & \quad x = 3t^2 \\
3. & \quad W = 2.1A + 5.2 \\
4. & \quad v = 11.11t + 6.67
\end{aligned}
}
\]