1. What is a Proportional Relationship?

Definition: A proportional relationship exists when two quantities change at a constant rate, meaning their ratio remains the same.

Key Equation:

\( y = kx \)

where \( k \) = constant of proportionality (also called constant ratio or unit rate)

Constant of Proportionality Formula:

\( k = \frac{y}{x} \)

Characteristics of Proportional Relationships:

• The ratio \( \frac{y}{x} \) is constant for all pairs of values
• The graph is a straight line that passes through the origin (0, 0)
• The equation has the form \( y = kx \) (no added constant)
• When \( x = 0 \), then \( y = 0 \)
• The constant \( k \) represents the slope of the line

2. Find the Constant of Proportionality from a Table

Steps:

1. Choose any pair of \( x \) and \( y \) values from the table
2. Divide \( y \) by \( x \): \( k = \frac{y}{x} \)
3. Verify by checking other pairs (they should all give the same \( k \))

Example:

Find the constant of proportionality from this table:

Solution:

Using first pair: \( k = \frac{y}{x} = \frac{8}{2} = 4 \)

Check with second pair: \( k = \frac{16}{4} = 4 \) ✓

Check with third pair: \( k = \frac{24}{6} = 4 \) ✓

The constant of proportionality is 4.

3. Write Equations for Proportional Relationships from Tables

Steps:

1. Find the constant of proportionality \( k = \frac{y}{x} \)
2. Write the equation in the form \( y = kx \)
3. Substitute the value of \( k \) into the equation

Example:

Write an equation for the relationship shown in this table:

Step 1: Find \( k \): \( k = \frac{12}{3} = 4 \)

Step 2: Write equation: \( y = kx \)

Step 3: Substitute: \( y = 4x \)

Equation: \( y = 4x \)

4. Identify Proportional Relationships by Graphing

A graph represents a proportional relationship if and only if:

Two Requirements:

1. The graph is a straight line (linear relationship)
2. The line passes through the origin (0, 0)

NOT Proportional If:

• The line does NOT pass through (0, 0)
• The graph is curved (not a straight line)
• The y-intercept is not zero

Examples:

Proportional: A line passing through (0,0), (1,3), (2,6), (3,9)

NOT Proportional: A line passing through (0,2), (1,5), (2,8) — doesn't pass through origin

5. Find the Constant of Proportionality from a Graph

Methods:

Method 1: Use any point on the line

1. Choose any point (x, y) on the line (not the origin)
2. Calculate \( k = \frac{y}{x} \)

Method 2: Use the slope formula

The constant of proportionality equals the slope: \( k = \text{slope} = \frac{\text{rise}}{\text{run}} \)

Example:

A proportional relationship graph passes through points (0, 0) and (4, 12). Find the constant of proportionality.

Using (4, 12): \( k = \frac{y}{x} = \frac{12}{4} = 3 \)

Using slope: From (0,0) to (4,12), rise = 12, run = 4

\( k = \frac{12}{4} = 3 \)

The constant of proportionality is 3.

6. Write Equations for Proportional Relationships from Graphs

Steps:

1. Find the constant of proportionality from the graph
2. Write the equation \( y = kx \)

Example:

A line passes through the origin and the point (2, 10). Write an equation.

Step 1: Find \( k \): \( k = \frac{10}{2} = 5 \)

Step 2: Write equation: \( y = 5x \)

7. Identify Proportional Relationships from Tables

Test for Proportionality:

Calculate \( \frac{y}{x} \) for each row. If all ratios are equal, the relationship is proportional.

Example 1: Proportional

✓ All ratios equal 3 → This IS proportional

Example 2: NOT Proportional

✗ Ratios are different → This is NOT proportional

8. Identify Proportional Relationships from Graphs and Equations

From Graphs:

Proportional: Straight line through the origin (0, 0)

NOT Proportional: Line doesn't pass through origin OR curved line

From Equations:

Proportional: Equation in form \( y = kx \) (where k is a constant)

Examples: \( y = 3x \), \( y = 0.5x \), \( y = \frac{2}{3}x \)

NOT Proportional: Equation has an added constant or different form

Examples: \( y = 3x + 2 \), \( y = x^2 \), \( y = 5 \)

Quick Check:

9. Identify Proportional Relationships: Word Problems

Steps to Solve:

1. Read the problem carefully and identify the two quantities
2. Check if the ratio between quantities is constant
3. Check if when one quantity is zero, the other is also zero
4. Find the constant of proportionality if it's proportional

Examples:

Example 1: Maria earns $12 per hour. Is the relationship between hours worked and money earned proportional?

Analysis: Each hour = $12

1 hour = $12, 2 hours = $24, 3 hours = $36

Ratio: \( \frac{12}{1} = \frac{24}{2} = \frac{36}{3} = 12 \) (constant)

0 hours = $0

✓ Yes, this is proportional with \( k = 12 \)

Equation: \( y = 12x \) (where y = money, x = hours)

Example 2: A taxi charges $5 plus $2 per mile. Is the relationship between miles and cost proportional?

Analysis: 0 miles = $5 (not $0!)

There's a flat fee, so the ratio is not constant

✗ No, this is NOT proportional

Equation: \( y = 2x + 5 \) (has a constant added)

10. Graph Proportional Relationships and Find the Slope

Key Concept: For proportional relationships, the constant of proportionality equals the slope of the line.

Slope Formula:

\( \text{Slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1} \)

For Proportional Relationships:

\( \text{Slope} = k = \frac{y}{x} \)

Steps to Graph:

1. Start at the origin (0, 0)
2. Use the constant \( k \) to find other points
3. Plot points and draw a straight line through them

Example:

Graph \( y = 3x \) and find the slope.

Points: (0, 0), (1, 3), (2, 6), (3, 9)

Slope: \( k = 3 \) (for every 1 unit right, go 3 units up)

Or using formula: \( \frac{3-0}{1-0} = 3 \)

11. Interpret Graphs of Proportional Relationships

What to Look For:

• Slope/Constant: Tells the rate of change (how much y changes per unit of x)
• Steepness: Higher slope = steeper line = faster rate of change
• Points on line: Can be used to find specific values
• Context: What do x and y represent in the real situation?

Example:

A graph shows the relationship between gallons of gas and cost. The line passes through (0,0) and (5, 20).

Interpretation:

• The constant of proportionality: \( k = \frac{20}{5} = 4 \)

• This means gas costs $4 per gallon

• Equation: \( y = 4x \) (where y = cost, x = gallons)

• For any number of gallons, multiply by 4 to find the cost

12. Write and Solve Equations for Proportional Relationships

Steps:

1. Identify the constant of proportionality from the problem
2. Write the equation \( y = kx \)
3. Substitute the known value to solve for the unknown

Examples:

Example 1: A car travels at a constant speed of 60 miles per hour. How far will it travel in 3.5 hours?

Step 1: \( k = 60 \) (miles per hour)

Step 2: Equation: \( y = 60x \) (y = distance, x = time)

Step 3: \( y = 60(3.5) = 210 \) miles

Example 2: A recipe uses 2 cups of flour for every 3 cups of sugar. If you use 15 cups of sugar, how many cups of flour do you need?

Step 1: \( k = \frac{2}{3} \) (flour per sugar)

Step 2: Equation: \( y = \frac{2}{3}x \) (y = flour, x = sugar)

Step 3: \( y = \frac{2}{3}(15) = 10 \) cups of flour

13. Compare Proportional Relationships Represented in Different Ways

Key Concept: Compare proportional relationships by comparing their constants of proportionality (k values).

How to Compare:

1. Find \( k \) for each relationship
2. The relationship with the larger \( k \) has a faster rate of change
3. The relationship with the larger \( k \) has a steeper graph

Example:

Compare these three proportional relationships:

Relationship A (Table):

\( k_A = \frac{10}{2} = 5 \)

Relationship B (Equation): \( y = 3x \)

\( k_B = 3 \)

Relationship C (Graph): Line passes through (0, 0) and (1, 7)

\( k_C = \frac{7}{1} = 7 \)

Comparison:

\( k_C > k_A > k_B \) → \( 7 > 5 > 3 \)

Order (fastest to slowest): Relationship C, Relationship A, Relationship B

Quick Reference: Proportional Relationships

Key Formula:

\( y = kx \) where \( k = \frac{y}{x} \)

Characteristics:

• Constant ratio: \( \frac{y}{x} \) is always the same
• Graph: Straight line through origin
• Equation: \( y = kx \) form (no constant added)
• When x = 0, then y = 0
• k = slope = constant of proportionality

How to Find k:

• From table: \( k = \frac{y}{x} \) for any pair
• From graph: Use any point or find slope
• From equation: k is the coefficient of x
• From word problem: Find the unit rate

💡 Key Tips for Proportional Relationships

• ✓ Proportional = y = kx (no added constant!)
• ✓ Graph must pass through (0, 0) to be proportional
• ✓ Constant of proportionality k = y/x for any point
• ✓ k is also the slope and the unit rate
• ✓ Check multiple ratios in tables to verify proportionality
• ✓ All ratios must be identical for proportional relationship
• ✓ Larger k = steeper line = faster rate
• ✓ When comparing: find each k value first
• ✓ y = 3x + 2 is NOT proportional (has +2)
• ✓ Word problems: look for constant rate or unit price
• ✓ If one quantity is 0, the other must also be 0
• ✓ Practice identifying proportional vs. non-proportional