Printable math worksheet for graphing proportional relationships.

The task asks you to draw graphs for the proportional relationship given by the equation:

$$
y = kx
$$

This is a direct proportion, where $ y $ is directly proportional to $ x $, and $ k $ is the constant of proportionality (also the slope of the line). The graph of $ y = kx $ is always a straight line passing through the origin (0, 0).

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🔧 How to Graph $ y = kx $:

1. Start at the origin: $ (0, 0) $
2. Use the value of $ k $ to determine the slope.
- For example, if $ k = 2 $, then for every 1 unit increase in $ x $, $ y $ increases by 2 units.
3. Plot a few points using the equation $ y = kx $, then draw a straight line through them.

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Let’s go through each case:

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✔ 1) $ k = 2 $ → $ y = 2x $

- Points:
- $ x = 0 $ → $ y = 0 $ → $ (0, 0) $
- $ x = 1 $ → $ y = 2 $ → $ (1, 2) $
- $ x = 2 $ → $ y = 4 $ → $ (2, 4) $
- $ x = -1 $ → $ y = -2 $ → $ (-1, -2) $

- Draw a straight line through these points on the grid.

> ✔️ This line will rise steeply (slope = 2).

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✔ 2) $ k = \frac{2}{9} $ → $ y = \frac{2}{9}x $

- Points:
- $ x = 0 $ → $ y = 0 $ → $ (0, 0) $
- $ x = 9 $ → $ y = 2 $ → $ (9, 2) $
- $ x = 18 $ → $ y = 4 $ → $ (18, 4) $
- $ x = -9 $ → $ y = -2 $ → $ (-9, -2) $

- Since $ k $ is small, the line will be very shallow (almost flat), rising slowly.

> ✔️ Use the grid carefully — scale matters!

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✔ 3) $ k = \frac{3}{4} $ → $ y = \frac{3}{4}x $

- Points:
- $ x = 0 $ → $ y = 0 $ → $ (0, 0) $
- $ x = 4 $ → $ y = 3 $ → $ (4, 3) $
- $ x = 8 $ → $ y = 6 $ → $ (8, 6) $
- $ x = -4 $ → $ y = -3 $ → $ (-4, -3) $

- This line has a moderate positive slope.

> ✔️ Connect these points with a straight line.

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✔ 4) $ k = 6 $ → $ y = 6x $

- Points:
- $ x = 0 $ → $ y = 0 $ → $ (0, 0) $
- $ x = 1 $ → $ y = 6 $ → $ (1, 6) $
- $ x = 2 $ → $ y = 12 $ → $ (2, 12) $
- $ x = -1 $ → $ y = -6 $ → $ (-1, -6) $

- This line is very steep — rises rapidly.

> ✔️ Be careful with the scale; make sure your graph fits within the grid.

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✔ 5) $ k = 1 $ → $ y = x $

- This is the identity function.
- Points:
- $ (0, 0), (1, 1), (2, 2), (-1, -1), \dots $

- The line goes through the origin at a 45° angle.

> ✔️ It's a diagonal line from bottom-left to top-right.

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✔ 6) $ k = \frac{5}{7} $ → $ y = \frac{5}{7}x $

- Points:
- $ x = 0 $ → $ y = 0 $ → $ (0, 0) $
- $ x = 7 $ → $ y = 5 $ → $ (7, 5) $
- $ x = 14 $ → $ y = 10 $ → $ (14, 10) $
- $ x = -7 $ → $ y = -5 $ → $ (-7, -5) $

- Moderate slope, less than 1.

> ✔️ Plot $ (7, 5) $ and connect back to origin.

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📌 Summary of Steps for Each Graph:

1. Start at (0, 0) — all lines pass through the origin.
2. Choose two or three values of $ x $ (positive and negative).
3. Calculate $ y = kx $ for each $ x $.
4. Plot the points.
5. Draw a straight line through them.

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💡 Tips:

- Use the grid lines carefully.
- If $ k $ is a fraction, pick an $ x $ that makes $ y $ a whole number (e.g., for $ k = \frac{2}{9} $, use $ x = 9 $).
- Label your axes if needed (though not required here).
- The steeper the line, the larger the $ k $.

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✔ Final Answer:
For each part, draw a straight line through the origin with slope equal to $ k $. Use the method above to plot key points and draw the line.

Let me know if you'd like me to generate the actual graph images or describe how to sketch them step-by-step!