Inverse Proportion Worksheet 2 featuring tables and graphs for various inverse proportion scenarios.

Let's solve each part of this Inverse Proportion Worksheet 2 step by step. The key concept is:

> If $ y $ is inversely proportional to $ x $, then:
> $$
> y = \frac{k}{x}
> $$
> where $ k $ is a constant.

Similarly:
- If $ y $ is inversely proportional to $ x^2 $, then: $ y = \frac{k}{x^2} $
- If $ y $ is inversely proportional to $ x^3 $, then: $ y = \frac{k}{x^3} $

We'll use the given values to find $ k $, then fill in the missing values in the tables.

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🔹 Problem 1: $ y \propto \frac{1}{x} $, and when $ x = 2 $, $ y = 6 $

So:
$$
y = \frac{k}{x} \Rightarrow 6 = \frac{k}{2} \Rightarrow k = 12
$$

Now compute $ y = \frac{12}{x} $ for each $ x $:

| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| y | 12 | 6 | 4 | 3 | 2.4 | 2 |

✔ So:
- $ y = 12, 6, 4, 3, 2.4, 2 $

---

🔹 Problem 2: $ y \propto \frac{1}{x} $, and when $ x = 3 $, $ y = 18 $

$$
18 = \frac{k}{3} \Rightarrow k = 54
\Rightarrow y = \frac{54}{x}
$$

| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| y | 54 | 27 | 18 | 13.5 | 10.8 | 9 |

✔ So:
- $ y = 54, 27, 18, 13.5, 10.8, 9 $

---

🔹 Problem 3: $ y \propto \frac{1}{x} $, and when $ x = 10 $, $ y = 35 $

$$
35 = \frac{k}{10} \Rightarrow k = 350
\Rightarrow y = \frac{350}{x}
$$

| x | 1 | 2 | 4 | 8 | 10 | 12 |
|---|---|---|---|---|----|----|
| y | 350 | 175 | 87.5 | 43.75 | 35 | 29.166... |

✔ So:
- $ y = 350, 175, 87.5, 43.75, 35, 29.17 $ (approx)

---

🔹 Problem 4: $ y \propto \frac{1}{x} $, and when $ x = 1 $, $ y = 8 $

$$
8 = \frac{k}{1} \Rightarrow k = 8
\Rightarrow y = \frac{8}{x}
$$

| x | -3 | -2 | -1 | 1 | 2 | 3 |
|---|----|----|----|---|---|---|
| y | -8/3 ≈ -2.67 | -4 | -8 | 8 | 4 | 8/3 ≈ 2.67 |

✔ So:
- $ y = -2.67, -4, -8, 8, 4, 2.67 $

---

🔹 Problem 5: $ y \propto \frac{1}{x} $, and when $ x = 2 $, $ y = 15 $

$$
15 = \frac{k}{2} \Rightarrow k = 30
\Rightarrow y = \frac{30}{x}
$$

| x | -6 | -4 | -2 | 2 | 4 | 6 |
|---|----|----|----|---|---|---|
| y | -5 | -7.5 | -15 | 15 | 7.5 | 5 |

✔ So:
- $ y = -5, -7.5, -15, 15, 7.5, 5 $

---

🔹 Problem 6: $ y \propto \frac{1}{x} $, and when $ x = 2 $, $ y = 10 $

$$
10 = \frac{k}{2} \Rightarrow k = 20
\Rightarrow y = \frac{20}{x}
$$

| x | -6 | -4 | -2 | 2 | 4 | 6 |
|---|----|----|----|---|---|---|
| y | -20/6 ≈ -3.33 | -5 | -10 | 10 | 5 | 3.33 |

✔ So:
- $ y = -3.33, -5, -10, 10, 5, 3.33 $

---

🔹 Problem 7: $ y \propto \frac{1}{x^2} $, and when $ x = 2 $, $ y = 9 $

$$
y = \frac{k}{x^2} \Rightarrow 9 = \frac{k}{4} \Rightarrow k = 36
\Rightarrow y = \frac{36}{x^2}
$$

| x | -3 | -2 | -1 | 1 | 2 | 3 |
|---|----|----|----|---|---|---|
| x² | 9 | 4 | 1 | 1 | 4 | 9 |
| y | 36/9 = 4 | 36/4 = 9 | 36/1 = 36 | 36/1 = 36 | 9 | 4 |

✔ So:
- $ y = 4, 9, 36, 36, 9, 4 $

---

🔹 Problem 8: $ y \propto \frac{1}{x^2} $, and when $ x = -2 $, $ y = \frac{16}{9} $

$$
\frac{16}{9} = \frac{k}{(-2)^2} = \frac{k}{4} \Rightarrow k = \frac{64}{9}
\Rightarrow y = \frac{64}{9x^2}
$$

Now compute:

| x | -6 | -4 | -2 | 2 | 4 | 6 |
|---|----|----|----|---|---|---|
| x² | 36 | 16 | 4 | 4 | 16 | 36 |
| y | $ \frac{64}{9 \cdot 36} = \frac{64}{324} = \frac{16}{81} $ | $ \frac{64}{9 \cdot 16} = \frac{64}{144} = \frac{4}{9} $ | $ \frac{64}{9 \cdot 4} = \frac{64}{36} = \frac{16}{9} $ | same as above | $ \frac{4}{9} $ | $ \frac{16}{81} $ |

✔ So:
- $ y = \frac{16}{81}, \frac{4}{9}, \frac{16}{9}, \frac{16}{9}, \frac{4}{9}, \frac{16}{81} $

---

🔹 Problem 9: $ y \propto \frac{1}{x^3} $, and when $ x = 2 $, $ y = 25 $

$$
25 = \frac{k}{(2)^3} = \frac{k}{8} \Rightarrow k = 200
\Rightarrow y = \frac{200}{x^3}
$$

| x | -6 | -4 | -2 | 2 | 4 | 6 |
|---|----|----|----|---|---|---|
| x³ | -216 | -64 | -8 | 8 | 64 | 216 |
| y | $ \frac{200}{-216} = -\frac{25}{27} $ | $ \frac{200}{-64} = -\frac{25}{8} = -3.125 $ | $ \frac{200}{-8} = -25 $ | $ \frac{200}{8} = 25 $ | $ \frac{200}{64} = \frac{25}{8} = 3.125 $ | $ \frac{200}{216} = \frac{25}{27} $ |

✔ So:
- $ y = -\frac{25}{27}, -3.125, -25, 25, 3.125, \frac{25}{27} $

---

## ✔ Final Answers (All Tables Filled):

1. $ y \propto \frac{1}{x} $, $ x=2 \Rightarrow y=6 $

| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| y | 12 | 6 | 4 | 3 | 2.4 | 2 |

2. $ y \propto \frac{1}{x} $, $ x=3 \Rightarrow y=18 $

| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| y | 54 | 27 | 18 | 13.5 | 10.8 | 9 |

3. $ y \propto \frac{1}{x} $, $ x=10 \Rightarrow y=35 $

| x | 1 | 2 | 4 | 8 | 10 | 12 |
|---|---|---|---|---|----|----|
| y | 350 | 175 | 87.5 | 43.75 | 35 | 29.17 |

4. $ y \propto \frac{1}{x} $, $ x=1 \Rightarrow y=8 $

| x | -3 | -2 | -1 | 1 | 2 | 3 |
|---|----|----|----|---|---|---|
| y | -2.67 | -4 | -8 | 8 | 4 | 2.67 |

5. $ y \propto \frac{1}{x} $, $ x=2 \Rightarrow y=15 $

| x | -6 | -4 | -2 | 2 | 4 | 6 |
|---|----|----|----|---|---|---|
| y | -5 | -7.5 | -15 | 15 | 7.5 | 5 |

6. $ y \propto \frac{1}{x} $, $ x=2 \Rightarrow y=10 $

| x | -6 | -4 | -2 | 2 | 4 | 6 |
|---|----|----|----|---|---|---|
| y | -3.33 | -5 | -10 | 10 | 5 | 3.33 |

7. $ y \propto \frac{1}{x^2} $, $ x=2 \Rightarrow y=9 $

| x | -3 | -2 | -1 | 1 | 2 | 3 |
|---|----|----|----|---|---|---|
| y | 4 | 9 | 36 | 36 | 9 | 4 |

8. $ y \propto \frac{1}{x^2} $, $ x=-2 \Rightarrow y=\frac{16}{9} $

| x | -6 | -4 | -2 | 2 | 4 | 6 |
|---|----|----|----|---|---|---|
| y | $ \frac{16}{81} $ | $ \frac{4}{9} $ | $ \frac{16}{9} $ | $ \frac{16}{9} $ | $ \frac{4}{9} $ | $ \frac{16}{81} $ |

9. $ y \propto \frac{1}{x^3} $, $ x=2 \Rightarrow y=25 $

| x | -6 | -4 | -2 | 2 | 4 | 6 |
|---|----|----|----|---|---|---|
| y | $ -\frac{25}{27} $ | $ -3.125 $ | $ -25 $ | $ 25 $ | $ 3.125 $ | $ \frac{25}{27} $ |

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## 📈 Graph Sketching Tips:

For inverse proportion ($ y = \frac{k}{x} $):
- Two branches: one in Q1 and Q3 if $ k > 0 $, or Q2 and Q4 if $ k < 0 $
- As $ x \to 0 $, $ y \to \pm\infty $
- As $ x \to \pm\infty $, $ y \to 0 $
- Symmetric about origin

For $ y \propto \frac{1}{x^2} $:
- Always positive (since $ x^2 > 0 $), symmetric about y-axis
- U-shaped on both sides, approaching zero as $ |x| \to \infty $
- Vertical asymptote at $ x = 0 $

For $ y \propto \frac{1}{x^3} $:
- Sign depends on sign of $ x $
- Odd function: symmetric about origin
- As $ x \to 0^+ $, $ y \to +\infty $; $ x \to 0^- $, $ y \to -\infty $
- As $ x \to \pm\infty $, $ y \to 0 $

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## ✔ Summary:

You now have all the values filled in the tables. To sketch the graphs:
1. Plot the points from the table.
2. Draw smooth curves through them, respecting asymptotes and symmetry.
3. Use arrows to show behavior near asymptotes.

Let me know if you'd like the graphs sketched (as text-based approximations) or want help plotting them!