Level Ups: 8th Grade Proportional Relationships | Made By Teachers - Free Printable

Let's solve each part of this worksheet step by step.

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Part 1: Graphing Equations

We are asked to graph three proportional relationships:

#### 1. $ y = 2x $

This is a direct proportion where $ y $ is twice the value of $ x $.
To graph it:
- When $ x = 0 $, $ y = 0 $
- When $ x = 1 $, $ y = 2 $
- When $ x = 2 $, $ y = 4 $
- When $ x = 3 $, $ y = 6 $
- And so on...

Plot these points and draw a straight line through them starting from the origin (0,0). The graph will be a straight line with a slope of 2.

✔ Graph: Plot points like (0,0), (1,2), (2,4), (3,6), etc., and connect them.

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#### 2. $ y = \frac{1}{4}x $

Here, $ y $ is one-fourth of $ x $.
- $ x = 0 $ → $ y = 0 $
- $ x = 4 $ → $ y = 1 $
- $ x = 8 $ → $ y = 2 $
- $ x = 12 $ → $ y = 3 $

Plot these and draw a straight line through the origin with a gentle slope.

✔ Graph: Points: (0,0), (4,1), (8,2), (12,3) — connect them.

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#### 3. $ y = \frac{2}{3}x $

$ y $ is two-thirds of $ x $.
- $ x = 0 $ → $ y = 0 $
- $ x = 3 $ → $ y = 2 $
- $ x = 6 $ → $ y = 4 $
- $ x = 9 $ → $ y = 6 $

Plot these points and draw a line through the origin.

✔ Graph: Points: (0,0), (3,2), (6,4), (9,6)

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> 💡 All three graphs are straight lines passing through the origin because they represent proportional relationships ($ y = kx $).

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Part 2: Write an Equation for Each Table

We need to find the constant of proportionality $ k $ such that $ y = kx $, using the given tables.

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#### 4.
| x | 0 | 3 | 6 | 9 | 12 |
|---|---|---|---|---|----|
| y | 0 | 45 | 90 | 135 | 180 |

Check ratio $ \frac{y}{x} $:
- $ \frac{45}{3} = 15 $
- $ \frac{90}{6} = 15 $
- $ \frac{135}{9} = 15 $
- $ \frac{180}{12} = 15 $

So, $ k = 15 $

✔ Equation: $ y = 15x $

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#### 5.
| x | 0 | 2 | 5 | 10 | 15 |
|---|---|---|---|----|----|
| y | 0 | 0.6 | 1.5 | 3 | 4.5 |

Compute $ \frac{y}{x} $:
- $ \frac{0.6}{2} = 0.3 $
- $ \frac{1.5}{5} = 0.3 $
- $ \frac{3}{10} = 0.3 $
- $ \frac{4.5}{15} = 0.3 $

So, $ k = 0.3 $

✔ Equation: $ y = 0.3x $

(Or $ y = \frac{3}{10}x $)

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#### 6.
| x | 0 | 10 | 24 | 50 | 100 |
|---|---|----|----|----|-----|
| y | 0 | 12 | 28.8 | 60 | 120 |

Check ratios:
- $ \frac{12}{10} = 1.2 $
- $ \frac{28.8}{24} = 1.2 $
- $ \frac{60}{50} = 1.2 $
- $ \frac{120}{100} = 1.2 $

So, $ k = 1.2 $

✔ Equation: $ y = 1.2x $

(Or $ y = \frac{6}{5}x $)

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#### 7.
| x | 0 | 6 | 12 | 18 | 60 |
|---|---|---|----|----|----|
| y | 0 | 8 | 16 | 24 | 80 |

Check ratios:
- $ \frac{8}{6} = \frac{4}{3} \approx 1.333 $
- $ \frac{16}{12} = \frac{4}{3} $
- $ \frac{24}{18} = \frac{4}{3} $
- $ \frac{80}{60} = \frac{4}{3} $

So, $ k = \frac{4}{3} $

✔ Equation: $ y = \frac{4}{3}x $

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✔ Final Answers:

#### Graphs (1–3):
- Plot points for each equation and draw straight lines through the origin.

#### Equations (4–7):
4. $ y = 15x $
5. $ y = 0.3x $ or $ y = \frac{3}{10}x $
6. $ y = 1.2x $ or $ y = \frac{6}{5}x $
7. $ y = \frac{4}{3}x $

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Let me know if you'd like the graphs sketched or printed!