Average Rate of Change: Graphs and Interpretation

Deck 2 of 2: Estimation, Interpretation, and Comparing Rates

In this deck:

• Estimate AROC from a graph using secant lines
• Interpret rates with units and compare across intervals

Learning Objectives for This Deck

For this deck, you should be able to:

4. Estimate AROC from a graph using approximate coordinates
5. Interpret AROC with appropriate units and meaning in context
6. Compare rates over intervals to identify trends and linearity

Estimating AROC from a Graph

When only a graph is given — no formula, no table:

1. Pick two points on the curve at the interval endpoints
2. Read their approximate coordinates and
3. Compute:

Approximate values are expected — the standard says "estimate."

Runner Graph: Two Secant Lines

Two intervals reveal different average speeds.

Reading the Runner: Interval [0, 4]

From the graph: and

The secant has slope 150 — the first four minutes were fast.

Reading the Runner: Interval [4, 8]

From the graph: and

The secant has slope 62.5 — the second interval was slower.

Estimation Is Expected — Use Approximately

When reading from a graph:

• Exact coordinates are rarely readable
• Different students may read slightly different values
• Both answers can be correct if the reading is reasonable

Use the ≈ symbol. Write: "approximately 62 m/min."

The process matters more than the exact number.

Negative Rate Means the Function Decreases

Stock price declines from to :

Negative sign → price is falling.

Reading: 120 dollars at , 90 dollars at .

Check: Which Interval Is Steeper?

Looking at the runner graph (Slide 4), compare the two secant lines.

• Which interval had the greater magnitude of AROC?
• What does the steeper secant mean physically?

Answer before the next slide.

Answer: First Interval Is Faster

• Interval [0, 4]: AROC = 150 m/min — steeper secant
• Interval [4, 8]: AROC = 62.5 m/min — less steep secant

Greater magnitude → faster average pace → steeper secant

Visual check: steeper secant always means greater magnitude.

Turning a Rate into a Meaningful Statement

A rate without context is incomplete.

Full interpretation requires three parts:

1. Number — the computed value
2. Units — output units per input unit
3. Meaning — what it says about the situation

Example: "The sunflower grew at 5.5 inches per week on average during weeks 2–6."

Population Table: Comparing All Five Rates

Are the rates constant or varying?

Varying rates → growth is non-linear and accelerating.

Testing for Linearity with Equal Intervals

If you compute AROC over consecutive equal-width intervals:

• Rates all the same → function is linear
• Rates vary → function is non-linear

For the population table: 1.2, 1.4, 1.8, 2.1, 2.6 thousand/year

Rates increase → non-linear, accelerating growth.

Overall Rate Can Hide Important Trends

Overall rate (weeks 0–10):

This single number hides the acceleration.

Interval rates reveal the true trend: 1.2, 1.4, 1.8, 2.1, 2.6

Units: The Key to Interpretation

For every additional mile per hour of speed, fuel efficiency increases 0.25 mpg.

The units output per input give the interpretation.

Check: Is the Population Linear?

From the table, the five 2-year rates are: 1.2, 1.4, 1.8, 2.1, 2.6 thousand/year.

1. Are these rates constant or changing?
2. Is the population function linear or non-linear?
3. What trend do the rates describe?

Write your three answers before the next slide.

Answer: Population Growth Is Non-Linear

1. Rates are increasing — not constant
2. Non-linear — rate of change varies with time
3. Trend: growth is accelerating — each 2-year period adds more people than the last

Overall: 1.82 thousand/year — but the per-interval picture is richer.

Key Takeaways from Deck Two

✓ Estimate from graphs: read coordinates, compute secant slope

✓ Approximations are expected — use ≈

✓ Interpret with: number + units + meaning

✓ Constant rates over equal intervals = linear function

Rate ≠ function value — "per" in units signals a rate

What Comes Next: Graphing Functions

HSF.IF.C.7 — Graphing Functions

• Graphing linear and quadratic functions with key features
• Understanding domain connects to graph extent
• AROC connects to slope of secant and tangent lines

Average rate of change is a foundational tool for all function analysis.