1. What is Rate of Change?

Definition: Rate of change is how much one quantity changes in relation to another quantity. It tells us how fast or slow something is changing.

Formula:

\( \text{Rate of Change} = \frac{\text{Change in Output}}{\text{Change in Input}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \)

Key Points:

• Positive rate: Output increases as input increases
• Negative rate: Output decreases as input increases
• Zero rate: Output stays the same (no change)
• Units: (units of y) per (units of x)

Real-World Examples:

• Speed: miles per hour (distance ÷ time)
• Price: dollars per item
• Growth: inches per year
• Wages: dollars per hour

2. Rate of Change: Tables

Steps to Find Rate of Change from a Table:

1. Choose two points from the table: \( (x_1, y_1) \) and \( (x_2, y_2) \)
2. Find the change in y: \( y_2 - y_1 \)
3. Find the change in x: \( x_2 - x_1 \)
4. Divide: \( \text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} \)
5. Interpret with units

Example 1: Constant Rate of Change

A car travels at a constant speed. Find the rate of change.

Using points (1, 60) and (3, 180):

\( \text{Rate of Change} = \frac{180 - 60}{3 - 1} = \frac{120}{2} = 60 \)

Interpretation: The car travels at 60 miles per hour.

Example 2: Variable Rate of Change

A plant's height over time. Find the average rate of change between day 0 and day 30.

Using points (0, 2) and (30, 14):

\( \text{Rate of Change} = \frac{14 - 2}{30 - 0} = \frac{12}{30} = 0.4 \)

Interpretation: The plant grows at an average rate of 0.4 inches per day.

3. Rate of Change: Graphs

Methods to Find Rate of Change from a Graph:

Method 1: Rise Over Run

1. Choose two points on the graph
2. Count the vertical change (rise): up is positive, down is negative
3. Count the horizontal change (run): right is positive, left is negative
4. Calculate: \( \text{Rate} = \frac{\text{rise}}{\text{run}} \)

Method 2: Use Coordinates

1. Identify coordinates of two points: \( (x_1, y_1) \) and \( (x_2, y_2) \)
2. Apply formula: \( \text{Rate} = \frac{y_2 - y_1}{x_2 - x_1} \)

Interpreting Graphs:

Example:

A graph shows temperature over time passing through points (2, 30) and (6, 50).

Rise: \( 50 - 30 = 20 \) degrees

Run: \( 6 - 2 = 4 \) hours

Rate of Change: \( \frac{20}{4} = 5 \) degrees per hour

Interpretation: Temperature increases by 5 degrees per hour.

4. Identify Graphs: Word Problems

Skill: Match a story or situation to its corresponding graph by understanding what the graph represents.

Key Graph Features to Look For:

1. Horizontal Line (Flat) → Something stays constant, no change

• Example: Standing still, pausing, resting, maintaining speed

2. Line Going Up (Positive Slope) → Something is increasing

• Steep: Fast increase (running, accelerating, rapid growth)
• Shallow: Slow increase (walking slowly, gradual growth)

3. Line Going Down (Negative Slope) → Something is decreasing

• Steep: Fast decrease (running downhill, draining quickly)
• Shallow: Slow decrease (walking down, slow drain)

4. Curved Line → Rate of change is varying

• Curve getting steeper: Accelerating, speeding up
• Curve getting flatter: Slowing down, decelerating

Steps to Match Story to Graph:

1. Read the story carefully and identify what's changing
2. Break the story into parts or time periods
3. For each part, determine if the quantity is:
   • Increasing (going up)
   • Decreasing (going down)
   • Staying constant (flat)
4. Determine if changes are fast (steep) or slow (shallow)
5. Match the pattern to the graph

Example 1: Height Over Time

Story: Sarah walks up a hill for 10 minutes, rests at the top for 5 minutes, then walks down for 10 minutes.

Graph sections:

• 0-10 minutes: Line going up (height increasing - walking uphill)
• 10-15 minutes: Horizontal line (height constant - resting at top)
• 15-25 minutes: Line going down (height decreasing - walking downhill)

Example 2: Distance from Home

Story: Mike drives away from home at a constant speed for 2 hours, stops for lunch for 30 minutes, then drives back home at a constant speed for 2 hours.

Graph sections:

• 0-2 hours: Straight line going up (distance from home increasing)
• 2-2.5 hours: Horizontal line (distance stays same - stopped)
• 2.5-4.5 hours: Straight line going down back to zero (returning home)

Example 3: Heart Rate During Exercise

Story: Jenny starts jogging and her heart rate increases rapidly. After 5 minutes, she runs at a steady pace and her heart rate stays elevated. After 20 minutes, she slows to a walk and her heart rate gradually decreases.

Graph sections:

• 0-5 minutes: Steep line going up (heart rate increasing rapidly)
• 5-20 minutes: Horizontal line at high level (heart rate elevated and steady)
• 20+ minutes: Line going down (heart rate decreasing gradually)

5. Common Graph Patterns and Their Meanings

6. Sketch and Describe Graphs

Describing a Graph:

When describing a graph, include:

• Variables: What does x represent? What does y represent?
• Overall trend: Is it increasing, decreasing, or staying constant?
• Rate of change: Is it changing fast or slow? Constant or variable?
• Key features: Flat sections, steep sections, turning points
• Starting and ending values: Where does it begin and end?

Sketching a Graph from a Description:

1. Label your axes with the variables and units
2. Identify the starting point
3. For each time period or section:
   • Draw a line going up (increasing)
   • Draw a line going down (decreasing)
   • Draw a horizontal line (constant)
4. Make sure the steepness matches the rate of change described
5. Check that all parts of the story are represented

Example Exercise:

Sketch a graph for: A balloon is inflated rapidly for 30 seconds, stays at maximum size for 10 seconds, then slowly deflates for 20 seconds.

Graph description:

• X-axis: Time (seconds)
• Y-axis: Balloon size (volume or diameter)
• 0-30 sec: Steep line going up (rapid inflation)
• 30-40 sec: Horizontal line at top (stays at maximum)
• 40-60 sec: Gentle line going down (slow deflation)

7. Understanding Context and Units

Why Context Matters:

The same graph shape can represent different situations depending on what the axes represent.

Examples of Different Contexts:

A line going up can mean:

• Distance from home increasing (traveling away)
• Temperature rising
• Water level in a pool increasing (filling up)
• Height of a plane increasing (ascending)
• Cost increasing with more items purchased

A horizontal line can mean:

• Car stopped (distance from home not changing)
• Temperature staying constant
• Bank balance not changing
• Heart rate steady

Always Include Units:

When stating rate of change, always include units to give meaning to the number.

• Good: 5 degrees per hour, 60 miles per hour, $10 per item
• Incomplete: Just "5" or "60" without units

Quick Reference: Interpret Functions

Rate of Change Formula:

\( \text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x} \)

Graph Interpretation Guide:

Key Steps:

• From table: Pick two points, use formula
• From graph: Find rise and run, divide
• Interpret: Always include units (per hour, per item, etc.)
• Match story: Look for increasing, decreasing, or constant sections

💡 Key Tips for Interpreting Functions

• ✓ Rate of change = slope = rise/run = Δy/Δx
• ✓ Always use TWO points to find rate of change
• ✓ Include units in your answer (per hour, per item, etc.)
• ✓ Positive rate = going up; Negative rate = going down
• ✓ Zero rate = horizontal line = no change
• ✓ Steep line = fast change; Shallow line = slow change
• ✓ Horizontal line = stopped, paused, resting, constant
• ✓ Line up = increasing (growing, speeding up, rising)
• ✓ Line down = decreasing (slowing, falling, draining)
• ✓ Break story into parts: what happens in each section?
• ✓ Check what x and y represent for context
• ✓ Curved graph = rate of change is varying