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To solve the problem of identifying the constant of proportionality for each graph, we need to determine the slope of each line. The constant of proportionality \( k \) in the equation \( y = kx \) is the slope of the line. Here's how we can find it for each graph:

General Steps:

1. Identify two points on the line: Typically, use the origin \((0, 0)\) and another point \((x_1, y_1)\) on the line.
2. Calculate the slope: The slope \( k \) is given by:
\[
k = \frac{y_1 - y_0}{x_1 - x_0}
\]
Since the line passes through the origin, \( (x_0, y_0) = (0, 0) \), the formula simplifies to:
\[
k = \frac{y_1}{x_1}
\]

Solutions for Each Graph:

#### 1)
- Points: \((0, 0)\) and \((15, 45)\)
- Slope:
\[
k = \frac{45 - 0}{15 - 0} = \frac{45}{15} = 3
\]
- Answer: \( y = 3x \)

#### 2)
- Points: \((0, 0)\) and \((24, 120)\)
- Slope:
\[
k = \frac{120 - 0}{24 - 0} = \frac{120}{24} = 5
\]
- Answer: \( y = 5x \)

#### 3)
- Points: \((0, 0)\) and \((25, 125)\)
- Slope:
\[
k = \frac{125 - 0}{25 - 0} = \frac{125}{25} = 5
\]
- Answer: \( y = 5x \)

#### 4)
- Points: \((0, 0)\) and \((5, 25)\)
- Slope:
\[
k = \frac{25 - 0}{5 - 0} = \frac{25}{5} = 5
\]
- Answer: \( y = 5x \)

#### 5)
- Points: \((0, 0)\) and \((10, 40)\)
- Slope:
\[
k = \frac{40 - 0}{10 - 0} = \frac{40}{10} = 4
\]
- Answer: \( y = 4x \)

#### 6)
- Points: \((0, 0)\) and \((35, 105)\)
- Slope:
\[
k = \frac{105 - 0}{35 - 0} = \frac{105}{35} = 3
\]
- Answer: \( y = 3x \)

#### 7)
- Points: \((0, 0)\) and \((40, 360)\)
- Slope:
\[
k = \frac{360 - 0}{40 - 0} = \frac{360}{40} = 9
\]
- Answer: \( y = 9x \)

#### 8)
- Points: \((0, 0)\) and \((32, 64)\)
- Slope:
\[
k = \frac{64 - 0}{32 - 0} = \frac{64}{32} = 2
\]
- Answer: \( y = 2x \)

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & y = 3x \\
2. & y = 5x \\
3. & y = 5x \\
4. & y = 5x \\
5. & y = 4x \\
6. & y = 3x \\
7. & y = 9x \\
8. & y = 2x \\
\end{array}
}
\]