A proportional relationship is a relationship where

two quantities maintain a CONSTANT RATIO

• As one quantity changes, the other changes at a constant rate

• The ratio y/x always equals the same value (k)

y = kx

where:

y = dependent variable

x = independent variable

k = constant of proportionality

✓ Graph: Straight line through the origin (0, 0)

✓ Table: All ratios y/x are equal

✓ Equation: Form y = kx (no constant added)

The constant of proportionality (k) is the

CONSTANT RATIO between two proportional quantities

• Also called the unit rate or rate of change

• Shows how much y changes for every 1 unit of x

k = y/x

or

k = y ÷ x

Step 1: Choose any pair of values (x, y) from the table

Step 2: Calculate k = y ÷ x

Step 3: Verify k is the same for all pairs (if proportional)

Table:

A table is proportional if:

✓ All ratios y/x are equal (constant k)

✓ When x = 0, y = 0 (passes through origin)

NOT proportional if:

✗ Ratios y/x are different

✗ When x = 0, y ≠ 0

Step 1: Find the constant of proportionality (k = y ÷ x)

Step 2: Write the equation: y = kx

Step 3: Substitute the value of k

From the previous table where k = 4:

A graph shows a proportional relationship if:

✓ It is a STRAIGHT LINE

✓ The line passes through the ORIGIN (0, 0)

NOT proportional if:

✗ The line is curved

✗ The line does NOT pass through the origin

✗ Example: y = 3x + 2 (does not pass through origin)

Step 1: Choose any point on the line (x, y)

Step 2: Calculate k = y ÷ x

Note: Do NOT use the origin (0, 0) for this calculation!

A line passes through (0, 0) and (4, 12). Find k.

For proportional relationships:

k = slope of the line

k = rise/run = change in y / change in x

Step 1: Find k from any point on the line (k = y/x)

Step 2: Write equation: y = kx

A line passes through (0, 0) and (3, 15). Write the equation.

Step 1: Identify what quantities are being compared

Step 2: Determine if relationship is proportional

Step 3: Find k (constant of proportionality)

Step 4: Write equation y = kx

Step 5: Use equation to solve problem

Problem: A car travels at a constant speed. In 2 hours, it travels 120 miles. How far will it travel in 5 hours?

Step 1: Find k from given values (k = y/x)

Step 2: Use equation y = kx

Step 3: Calculate missing values

Step 1: Plot the origin (0, 0)

Step 2: Plot points from the table

Step 3: Draw a straight line through all points

Step 4: Extend the line in both directions

✓ Equation form: Must be y = kx (no constant added or subtracted)

✓ Graph test: Line MUST pass through origin (0, 0)

✓ Table test: All ratios y/x must equal the same value

✓ Finding k: Divide y by x for any ordered pair

✓ k represents: The unit rate or rate of change

✓ When x = 0: y must also equal 0 in proportional relationships

✓ Slope = k: In proportional relationships, slope equals k

✓ Not proportional: y = 3x + 2 (has +2, doesn't pass through origin)

✓ k value: Can be any number (whole, decimal, or fraction)

✓ Real-world: Speed, cost per item, recipes are often proportional

Proportional Equation:

"Y equals K times X - that's the proportional way to flex!"

Finding k:

"To find k, divide y by x - that's the proportional mix!"

Graph Check:

"Through the origin it must go - straight line tells you so!"

Table Check:

"Same ratio every time - proportional relationship is prime!"

Equation Check:

"Y equals K times X, no adding allowed - that's the rule, don't be wowed!"

What k Means:

"K is the constant rate - how much y grows when x does wait!"

Origin Rule:

"Zero in, zero out - proportional without a doubt!"