DAT Quantitative Reasoning: Reading Line Graphs and Trends Over Time

DAT High-Yield Concept

Reading Values: Locate a data point by tracing vertically down to the horizontal axis (independent variable) and horizontally across to the vertical axis (dependent variable). Interpolate proportionally when points fall between gridlines.
Segment Trends: Positive slopes represent an increase, negative slopes represent a decrease, and flat horizontal lines signify no change between consecutive points.
Rate of Change: The line segment with the steepest slope (largest magnitude of $\frac{\Delta y}{\Delta x}$ ) carries the largest rate of change per unit of the horizontal axis.
Percent Change: Always use $\text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%$ . Divide by the initial (old) value, not the final value.

Reading Values and Gridline Interpolation

Line graphs display how a dependent variable changes over continuous ordered intervals, such as time. To read a specific coordinate pair $(x, y)$ , trace straight down from the plotted point to identify the horizontal axis value, and trace straight left to read the vertical axis value. When a plotted point falls between labeled gridlines, perform linear interpolation by determining the proportional position of the point between the surrounding grid values.

Reading a line graph: trace to the axis, then judge the slope — To read a value, trace from the plotted point down to the horizontal axis and across to the vertical axis, estimating the fraction of the gap when the point falls between gridlines. The May point lies three quarters of the way from 60 to 80, so it reads 75. Each segment is separately an increase, a decrease, or no change, and the steepest segment carries the largest change per month: March to April gains 30 patients, more than any other segment. Percent change is the new value minus the old value, divided by the old value, times one hundred, so January to April is 125 percent.

Worked Example: Interpolating Points Between Gridlines

Problem: On a line graph tracking bacterial culture density over time, the vertical axis has gridlines at 0.40 AU and 0.50 AU. At Hour 4, the plotted data point sits exactly halfway between these two gridlines. What is the estimated absorbance at Hour 4?

Solution:

Identify the lower gridline: $y_{1} = 0.40\text{ AU}$ .
Identify the upper gridline: $y_{2} = 0.50\text{ AU}$ .
Calculate the full gridline interval: $\Delta y = 0.50 - 0.40 = 0.10\text{ AU}$ .
Interpolate at the halfway mark ( $0.5$ ): $y = 0.40 + (0.5 \times 0.10) = 0.40 + 0.05 = 0.45\text{ AU}$ .

Common Exam Trap: Misreading the scale increment per gridline. Always calculate the exact numerical difference between adjacent gridlines before interpolating.

Describing Trends and Rates of Change

Line graphs connect consecutive data points with line segments to highlight trends over time. The direction and steepness (slope) of each segment convey how rapidly the variable changes:

Increase: Upward sloped segment ( $\Delta y > 0$ ).
Decrease: Downward sloped segment ( $\Delta y < 0$ ).
No Change: Flat horizontal segment ( $\Delta y = 0$ ).

The average rate of change per unit of the horizontal axis is given by the slope formula:

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

The steepest segment carries the largest rate of change per unit of the horizontal axis. While side-by-side grouped bar charts (covered in depth in its own lesson) display discrete categorical comparisons, line charts excel at illustrating continuous trends and volatility across sequential time intervals.

A line graph on the left and a bar graph on the right showing, line graphs are often the best means to present a quick overview of how two data sets vary in relation to one another than bar charts. (Source: Amrutha.alur · CC BY-SA 4.0 · Wikimedia Commons)

Segment Characteristics	Slope ( $m$ )	Trend Interpretation	Rate of Change Significance
Upward Sloping	$m > 0$	Increase between points	Positive growth rate
Downward Sloping	$m < 0$	Decrease between points	Negative growth rate (decline)
Horizontal Line	$m = 0$	No change (constant)	Zero rate of change
Steepest Line Segment	Largest $\|m\|$	Rapid rise or fall	Maximum rate of change per unit $x$

Calculating Percent Change Between Plotted Points

Percent Change Formula

To calculate relative percentage change between two plotted points:

\text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%

Mnemonic: Remember "NOO" for Percent Change: (New minus Old) divided by Old!

Worked Example: Rate of Change and Percent Change Analysis

Problem: A dental clinic records patient visits: Month 1 = 40, Month 2 = 50, Month 3 = 80, Month 4 = 80. Which 1-month interval experienced the steepest rate of change, and what was the percent change during that interval?

Solution:

Calculate rates of change ( $\frac{\Delta y}{\Delta x}$ $\frac{Δ y}{Δ x}$ ) for each interval:
- Month 1 to 2: $\frac{50 - 40}{2 - 1} = 10\text{ patients/month}$
- Month 2 to 3: $\frac{80 - 50}{3 - 2} = 30\text{ patients/month}$
- Month 3 to 4: $\frac{80 - 80}{4 - 3} = 0\text{ patients/month}$
Identify the steepest segment: Month 2 to 3 has the highest rate of change ( $30\text{ patients/month}$ ).
Calculate percent change from Month 2 ( $\text{Old Value} = 50$ ) to Month 3 ( $\text{New Value} = 80$ ): $\text{Percent Change} = \frac{80 - 50}{50} \times 100\% = \frac{30}{50} \times 100\% = 60\%$

Common Exam Trap: Erroneously dividing by the ending value ( $80$ ) instead of the initial value ( $50$ ), which yields an incorrect result of $37.5\%$ .

Critical Thinking Questions

If a quantity decreases from 200 to 150 in Period 1 and then increases from 150 back to 200 in Period 2, why are the percent changes different in magnitude even though the absolute change is 50 units in both cases?
How does modifying the scale increment on the vertical axis alter the visual steepness of a segment without changing its calculated rate of change?
In a multi-line graph comparing two dental treatments over time, what does the point where two lines intersect signify mathematically?

References

Sources returned by Google Search grounding during lesson verification.

2.3: Rates of Change and Behavior of Graphs - Mathematics LibreTexts — math.libretexts.org
Line Graphs to Display Data Over Time | CK-12 Foundation — ck12.org

Reading Line Graphs and Trends Over Time

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Practice Questions

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