EQN-05-11

Data Trends & Rates of Change

Read growth, decline, and the steepest change from a line graph, and compute percent change over time.

~5 min read•Difficulty 2/5

Reading a trend

A line graph rises when the quantity grows, falls when it shrinks, and is flat when it is unchanged. The steepest segment marks the fastest change — up or down.

Rate of change between two points

rate of change = (later value - earlier value) / (time elapsed)

This is exactly the slope of the line between the two points.

Largest change $\neq =$ largest value

The biggest single increase is the largest jump between consecutive points, not the tallest point. Compute consecutive differences.

Percent change over time

percent change = (new - old) / old \times 100

"Grew by what percent from year 1 to year 3" divides the rise by the year-1 value, not the latest.

Interpolation vs extrapolation

Reading a value between known points (interpolation) is fairly safe; predicting beyond the data (extrapolation) assumes the trend continues — flag it as an assumption.

Diagram

The steepest upward segment is the largest increase.

Worked examples

Example 1 (difficulty 2/5)

Yearly sales: 200, 260, 250, 400. Between which consecutive years was the largest increase?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

A value rose from 80 in Year 1 to 100 in Year 3. What is the percent increase?

◎ Predict the answer first:

Tips

The steepest segment is the fastest change; a falling segment is a decrease.
Largest increase $=$ biggest consecutive difference, not the tallest point.
Percent change always divides by the earlier (old) value.

Common mistakes

Calling the highest point the "biggest increase"

Why: Height grabs attention.

Fix: An increase is a difference between points; compare the steps.

Dividing by the new value for percent change

Why: The latest value is in front of you.

Fix: The base is the earlier value; divide the change by it.

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