Calculate and Interpret Average Rate of Change

Deck 1 of 2: Definition and Computation

In this deck:

• Connect AROC to the slope formula
• Compute AROC from symbolic expressions and tables

What You Will Learn Today

1. Calculate AROC over an interval:
2. Connect AROC to the slope of the secant line
3. Compute AROC from formulas and tables
4. Estimate AROC from a graph
5. Interpret AROC in context with appropriate units
6. Compare rates across intervals to identify trends

Two Real-World Rates — You Already Know How

Scenario A: A car travels 120 miles in 2 hours.

Scenario B: A rocket gains 2500 feet of altitude in 10 seconds.

What you computed: change in output ÷ change in input

The Average Rate of Change Formula

This equals the slope of the secant line from to .

Linear Functions Always Have Constant Rate

— plumber's cost in dollars

Over [1, 3]:

Over [0, 5]:

Both give 40 — equal to the slope.

Non-Linear Functions Have Varying Rate

The Secant Line — Visual Connection

The secant line connecting and :

• Slope = AROC =
• Steeper secant → faster average change
• For linear functions: secant IS the function line (same slope everywhere)

The secant line gives the "straight-line summary" of curved behavior.

Check: Is f(x) = 3x + 7 Linear?

For , compute the AROC over and .

Are the two results the same or different?

What does your answer tell you about ?

Compute before the next slide.

Answer: Both Intervals Give Slope 3

Over [1, 4]:

Over [0, 10]:

Conclusion: Same rate for any interval → is linear with slope 3.

Three Steps to Compute AROC

1. Evaluate and
2. Subtract: (ending minus starting)
3. Divide by (same order)

Include units: output units per input unit.

Subtract in the same direction in both numerator and denominator.

Symbolic Example: Zero Average Rate

— ball height in feet

Evaluate: and

Both heights equal 52 ft — surprising result!

Zero Rate Means Net Change Is Zero

AROC = 0 means the function ended where it started.

• Ball at and : same height (52 ft)
• Ball went UP then came back DOWN
• AROC = 0 captures net change, not total path

Drive 30 miles out and back → displacement = 0.

AROC from a Table of Values

Weeks 0 to 10: in/week

Weeks 4 to 8: in/week

The middle period grew faster than the overall average.

Uneven Spacing Works Just as Well

From to :

Check: Sunflower Rate, Weeks 2 to 6

Using this table:

Compute the average rate of change from week 2 to week 6.

Show all three steps, include units.

Check Answer: AROC from Week 2 to 6

Evaluate: ,

Average growth: 5.5 inches per week between weeks 2 and 6.

Key Takeaways from Deck One

✓ AROC = — slope formula generalized

✓ Equals the secant line slope between two points

✓ Linear → constant AROC; Non-linear → AROC varies

AROC = 0 means net change = 0, not a constant function

AROC is a rate (output/input), not an average output

Coming Up in Deck Two

Estimating AROC from Graphs and Interpreting in Context

• Reading approximate coordinates and computing slope
• The secant line as a visual tool
• Interpreting units: "output units per input unit"
• Comparing rates across intervals to detect trends

Applies LOs 4, 5, and 6