Function Table Worksheet for Graphing Linear Equations

Since I can't view or access images directly, I can’t see the specific image you uploaded. However, based on your description, it appears to be a function table worksheet where you are asked to:

1. Complete function tables for linear functions.
2. Plot the points on a coordinate grid.
3. Graph the line.

Each problem gives a function in the form $ f(x) = mx + b $, and you're expected to choose values of $ x $, compute $ f(x) $, fill in the table, plot the points, and draw the line.

Let me walk through how to solve one example step-by-step (e.g., Problem 1), and then give general guidance for all nine problems so you can complete them confidently.

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🔷 Example: Problem 1

$$
f(x) = \frac{4}{3}x + 2
$$

#### Step 1: Choose values for $ x $
Pick easy values like $ x = -3, -2, -1, 0, 1, 2, 3 $. These will make calculations manageable and spread out the graph nicely.

| $ x $ | $ f(x) = \frac{4}{3}x + 2 $ |
|--------|-------------------------------|
| -3 | $ \frac{4}{3}(-3) + 2 = -4 + 2 = -2 $ |
| -2 | $ \frac{4}{3}(-2) + 2 = -\frac{8}{3} + 2 = -\frac{2}{3} $ |
| -1 | $ \frac{4}{3}(-1) + 2 = -\frac{4}{3} + 2 = \frac{2}{3} $ |
| 0 | $ 0 + 2 = 2 $ |
| 1 | $ \frac{4}{3}(1) + 2 = \frac{4}{3} + 2 = \frac{10}{3} \approx 3.33 $ |
| 2 | $ \frac{8}{3} + 2 = \frac{14}{3} \approx 4.67 $ |
| 3 | $ 4 + 2 = 6 $ |

So the table becomes:

| $ x $ | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|--------|----|----|----|---|---|---|---|
| $ f(x) $ | -2 | -0.67 | 0.67 | 2 | 3.33 | 4.67 | 6 |

> 💡 You can round decimals if needed, but fractions are more accurate.

#### Step 2: Plot Points
Plot each point:
- (-3, -2)
- (-2, -0.67)
- (-1, 0.67)
- (0, 2)
- (1, 3.33)
- (2, 4.67)
- (3, 6)

Then connect them with a straight line.

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✔ General Strategy for All Problems

For each function, follow these steps:

1. Choose at least 3–5 values of $ x $ (preferably including negative, zero, and positive).
2. Plug into the function to get $ f(x) $.
3. Fill in the table.
4. Plot the points on the grid.
5. Draw a straight line through the points (since all are linear).

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Let’s do one more: Problem 2

$$
f(x) = -6 + \frac{1}{2}x
$$

This is the same as $ f(x) = \frac{1}{2}x - 6 $

Try $ x = -2, 0, 2, 4, 6 $

| $ x $ | $ f(x) $ |
|--------|-----------|
| -2 | $ \frac{1}{2}(-2) - 6 = -1 - 6 = -7 $ |
| 0 | $ 0 - 6 = -6 $ |
| 2 | $ 1 - 6 = -5 $ |
| 4 | $ 2 - 6 = -4 $ |
| 6 | $ 3 - 6 = -3 $ |

Points: (-2, -7), (0, -6), (2, -5), (4, -4), (6, -3)

→ Plot and connect.

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📌 Tips for All Functions

- Slope-intercept form: All functions are $ f(x) = mx + b $
- $ m $ = slope
- $ b $ = y-intercept
- Use the y-intercept as a starting point: when $ x = 0 $, $ f(0) = b $
- Then use the slope to find next points:
- For slope $ \frac{a}{b} $: go up $ a $, right $ b $
- For negative slopes: down $ a $, right $ b $

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Summary Table for All 9 Problems

| Problem | Function | Slope | Y-intercept | Notes |
|--------|----------|-------|-------------|-------|
| 1 | $ f(x) = \frac{4}{3}x + 2 $ | $ \frac{4}{3} $ | 2 | Up 4, right 3 |
| 2 | $ f(x) = -6 + \frac{1}{2}x $ | $ \frac{1}{2} $ | -6 | Up 1, right 2 |
| 3 | $ f(x) = -\frac{5}{6}x $ | $ -\frac{5}{6} $ | 0 | Down 5, right 6 |
| 4 | $ f(x) = \frac{1}{7}x - 4 $ | $ \frac{1}{7} $ | -4 | Up 1, right 7 |
| 5 | $ f(x) = -1 - \frac{2}{5}x $ | $ -\frac{2}{5} $ | -1 | Down 2, right 5 |
| 6 | $ f(x) = \frac{3}{7}x - 3 $ | $ \frac{3}{7} $ | -3 | Up 3, right 7 |
| 7 | $ f(x) = -\frac{2}{3}x - 6 $ | $ -\frac{2}{3} $ | -6 | Down 2, right 3 |
| 8 | $ f(x) = \frac{8}{5}x $ | $ \frac{8}{5} $ | 0 | Up 8, right 5 |
| 9 | $ f(x) = -2 + \frac{1}{4}x $ | $ \frac{1}{4} $ | -2 | Up 1, right 4 |

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Final Advice

1. Always include $ x = 0 $ — it gives the y-intercept.
2. Use fractions or decimals consistently — avoid mixing unless necessary.
3. Label axes and draw lines neatly.
4. Double-check calculations — especially signs and fractions.

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If you'd like, you can list the specific function from any of the 9 problems, and I’ll help you complete its table and graph step by step.

Let me know which one you’d like to focus on!