Direct Variation worksheet featuring graphs and tables to determine direct variation.

It looks like you've uploaded a worksheet titled "Direct Variation" that includes several problems related to identifying and working with direct variation relationships. However, since I can't view or access images directly, I can’t see the specific content of the image.

But based on the description and typical structure of such worksheets, here's how we can approach solving this kind of problem:

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🔍 What is Direct Variation?

A direct variation is a relationship between two variables where one variable is a constant multiple of the other. It follows the form:

$$
y = kx
$$

Where:
- $ y $ and $ x $ are variables,
- $ k $ is the constant of variation (a non-zero constant),
- The graph is a straight line passing through the origin (0, 0).

So, in a table or graph:
- If $ y $ increases proportionally as $ x $ increases,
- And the ratio $ \frac{y}{x} $ is constant for all pairs,
- Then it’s a direct variation.

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🧩 Typical Problems on This Worksheet

The worksheet likely includes:

1. Graphs – You're asked to identify which graphs show direct variation.
- ✔ A direct variation graph must be a straight line through the origin (0,0).
- ✘ Graphs that don’t pass through the origin or are curved are not direct variation.

2. Tables of values – You're given tables of $ x $ and $ y $ values.
- For each table, check if $ \frac{y}{x} $ is constant.
- If yes → direct variation → write equation $ y = kx $.
- If no → not a direct variation.

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✔ Step-by-Step Example

Let’s say one table is:

| x | y |
|---|---|
| 1 | 3 |
| 2 | 6 |
| 3 | 9 |

Check ratios:
- $ \frac{3}{1} = 3 $
- $ \frac{6}{2} = 3 $
- $ \frac{9}{3} = 3 $

✔ Constant ratio → direct variation.

Equation: $ y = 3x $

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Now, let’s suppose another table:

| x | y |
|---|---|
| 1 | 4 |
| 2 | 7 |
| 3 | 10 |

Ratios:
- $ \frac{4}{1} = 4 $
- $ \frac{7}{2} = 3.5 $
- $ \frac{10}{3} \approx 3.33 $

✘ Not constant → not direct variation.

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🔎 How to Solve the Worksheet

Here’s what you should do for each part:

#### Part 1: Which graph(s) show direct variation?
- Look at each graph.
- Check if it’s a straight line going through the origin (0,0).
- Only those qualify.

#### Part 2: Tables
For each table:
1. Compute $ \frac{y}{x} $ for each row.
2. If all values are equal → direct variation.
3. Write $ y = kx $, where $ k = \frac{y}{x} $.
4. If not constant → "No direct variation."

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📝 Sample Answer Format

For example, if Table 1 has:
| x | y |
|---|---|
| 2 | 6 |
| 4 | 12 |
| 6 | 18 |

Then:
- $ \frac{6}{2} = 3 $
- $ \frac{12}{4} = 3 $
- $ \frac{18}{6} = 3 $

Answer: Yes, direct variation; equation: $ y = 3x $

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