Line graphs and scatter plots — basic interpretation: Worked examples

This page is the "no-surprises" drill for the parent topic. We march through every kind of situation a graph can throw at you — rising, falling, flat, zero, negative x, an outlier, a trap about scale, a real-world word problem, a vertical (undefined-slope) edge case, and an exam twist — so that when you meet one in a test, you have already seen its twin.

Before any symbol appears we build it. If you have never plotted a point, start here: a point $(x, y)$ is just an instruction — "walk $x$ steps right, then $y$ steps up, and drop a dot." (If $x$ is negative you walk left; if $y$ is negative you go down.) That is the only prerequisite.

The scenario matrix

Every question about these graphs falls into one of the cells below. Each worked example is tagged with the cell(s) it covers.

Cell	What makes it special	Covered by
A. Rising line	y-values go up as x goes right	Example 1
B. Falling line	y-values go down	Example 2
C. Flat / constant	horizontal line, zero change	Example 2
D. Fluctuating + max/min/range	up-and-down, find peak & valley	Example 3
E. Negative y values	y dips below zero	Example 4
F. Rate comparison (Δy/Δx)	which interval is steeper	Example 5
G. Positive correlation	dots trend up	Example 6
H. Negative correlation	dots trend down	Example 7
I. Zero / no correlation	random cloud, degenerate case	Example 7
J. Outlier	one dot breaks the pattern	Example 8
K. Real-world word problem + prediction	interpolate to forecast	Example 9
L. Exam twist (scale trap + extrapolation)	the deliberate trick	Example 10
M. Negative x + vertical line (Δx = 0)	left of the y-axis; undefined slope	Example 11

Intuition Words and symbols you must own before we start

$Δ$ (the Greek letter delta) is shorthand for "the change in". So $Δ y$ means "how much y changed" — you get it by subtracting: $Δ y = y_{end} - y_{start}$ . It is not multiplication, it is a difference.
Rate of change $= \frac{Δ y}{Δ x}$ = "how many y-units per one x-unit". Picture a staircase: $Δ y$ is how tall one flight is, $Δ x$ is how far along you walked. The ratio tells you the steepness.
The straight-line formula is $y = m x + c$ . Here $m$ (the slope) is exactly that rate $\frac{Δ y}{Δ x}$ , and $c$ (the y-intercept) is the y-value where the line crosses $x = 0$ — its starting height. Both come from linear equations; we use each of them below.
Warning (Δx = 0): the rate $\frac{Δ y}{Δ x}$ needs $Δ x \neq = 0$ . If two points share the same x (a vertical line), then $Δ x = 0$ and you would be dividing by zero — the slope is then undefined, not zero. We meet this in Example 11.

Cell A — the rising line

Worked example Example 1 · Plant height (increasing trend)

A seedling is measured each week: Week 1 → 2 cm, Week 2 → 4 cm, Week 3 → 7 cm, Week 4 → 11 cm. Question: Is the plant growing, and by how much per week on average across these 4 weeks?

Forecast: Guess now — up or down? By roughly how many cm each week?

Plot the points $(1, 2), (2, 4), (3, 7), (4, 11)$ and join them. Why this step? You cannot see a trend from a table; the eye reads direction from a picture. Look at the orange line in the figure — every next point sits higher.
Figure 1 — Line graph of plant height rising from (1,2) up to (4,11); an orange upward-sloping line labelled "rising = increasing trend".
Name the direction: each y-value is larger than the one before → increasing trend. Why this step? "Rising line" is exactly the definition of an increasing trend; naming it lets us answer "is it growing?" with a confident yes.
Average rate $= \frac{Δ y}{Δ x} = \frac{11 - 2}{4 - 1} = \frac{9}{3} = 3$ cm/week. Why this step? "How fast" needs a number, not just "up". We divide total rise by total run — the staircase idea.

Verify: Start at 2 cm, add 3 cm/week for 3 weeks → $2 + 3 \times 3 = 11$ cm. Matches Week 4. Units: $\frac{cm}{week}$ ✓.

Cells B & C — the falling line and the flat line

Worked example Example 2 · Battery charge (decreasing then constant)

A phone's battery reads: Hour 0 → 100%, Hour 1 → 80%, Hour 2 → 60%, Hour 3 → 60%, Hour 4 → 60%. Question: Describe each part of the graph.

Forecast: Two behaviours are hiding here — can you spot where the line stops falling?

Plot $(0, 100), (1, 80), (2, 60), (3, 60), (4, 60)$ . Why this step? We need the shape before we describe it.
Figure 2 — Battery-percent line graph: a teal segment falling from (0,100) to (2,60), then a plum horizontal segment holding at 60 from hour 2 to hour 4.
Hours 0→2: y falls $100 \to 60$ → decreasing trend. Rate $= \frac{60 - 100}{2 - 0} = \frac{- 40}{2} = - 20$ %/hour. Why this step? A negative rate is the signature of a falling line — the sign itself tells the story. The teal segment tilts down.
Hours 2→4: y stays at 60 → constant. Rate $= \frac{60 - 60}{4 - 2} = \frac{0}{2} = 0$ . Why this step? A zero rate is the degenerate "flat" case — no change at all. The plum segment is perfectly horizontal.

Verify: Sign check — dropping charge must give a negative rate (it did, $- 20$ ). Flat part must give exactly zero (it did). ✓

Cell D — fluctuating: maximum, minimum, range

Cell E — the line that goes negative

Worked example Example 4 · Temperature crossing zero

Overnight temperatures (°C): 10 PM → 4, 11 PM → 1, 12 AM → $- 2$ , 1 AM → $- 5$ . Question: By how much did the temperature fall, and does it ever go below zero?

Forecast: Negative numbers are just "below the freezing line" — will the subtraction still work?

Plot, treating the x-axis (y = 0) as the freezing line. Points below it are negative. Why this step? Beginners panic at negatives, but on a graph a negative y is simply below the horizontal axis — nothing new.
Figure 3 — Overnight temperature line graph crossing a dashed teal "freezing line" at y = 0; points fall from (10PM, 4) to (1AM, −5), with the last two points sitting below zero.
Total change $= y_{end} - y_{start} = (- 5) - 4 = - 9$ °C. Why this step? The subtraction rule does not change for negatives. Subtracting a bigger start from a smaller (negative) end gives a negative $Δ y$ = "it fell 9 degrees".
Below zero? From midnight onward y is negative → yes, below freezing after 12 AM. Why this step? The sign of the y-value directly answers the "below zero" question.

Verify: Start 4, drop 9 → $4 - 9 = - 5$ = end value ✓. A fall gives a negative $Δ y$ ✓.

Cell F — comparing rates across intervals

Worked example Example 5 · Which period grew faster?

Revenue (₹ lakh, i.e. hundred-thousand rupees): Month 1 → 10, Month 4 → 19, Month 8 → 43. Question: In which interval did revenue grow faster, Months 1–4 or 4–8?

Forecast: Both go up — but "up faster" means steeper, not higher.

Interval 1–4: $\frac{Δ y}{Δ x} = \frac{19 - 10}{4 - 1} = \frac{9}{3} = 3$ ₹lakh/month. Why this step? Rate isolates speed of change; we must compute it separately for each interval to compare fairly.
Interval 4–8: $\frac{43 - 19}{8 - 4} = \frac{24}{4} = 6$ ₹lakh/month. Why this step? Same formula, second staircase flight.
Compare: $6 > 3$ → Months 4–8 grew faster (steeper line). Why this step? A larger $\frac{Δ y}{Δ x}$ = a steeper slope = faster growth. This is the slope of a graph speaking.

Verify: Rebuild: $10 + 3 (3) = 19$ ✓ and $19 + 6 (4) = 43$ ✓. Both rates positive (both rising) ✓.

Cell G — positive correlation on a scatter plot

Worked example Example 6 · Temperature vs. ice-cream sales

Points (°C, ₹ rupees): $(20, 300), (22, 400), (25, 500), (28, 650), (31, 900)$ . Question: What kind of correlation, and roughly what would sales be at 27 °C?

Forecast: Warmer days, more ice cream — up or down cloud?

Plot each dot — one dot = one day, x = temperature, y = sales. Why this step? A scatter plot's whole meaning is "each dot is one paired observation." See the figure.
Figure 4 — Scatter plot of temperature (x) vs. ice-cream sales in rupees (y); orange dots climb left-to-right with a dashed teal trend line, and a plum diamond marks the predicted ≈₹600 at 27 °C.
Read the pattern: as x increases the dots climb → positive correlation, and they hug a line closely → strong. Why this step? Direction (up) names the correlation type; tightness names the strength.
Predict at 27 °C (interpolation): 27 sits between 25 (₹500) and 28 (₹650). Halfway-ish → about ₹600. Why this step? Interpolation = reading between known points along the trend, which is safe because 27 lies inside our data range.

Verify: 27 is between 25 and 28, so the answer must be between ₹500 and ₹650 — and ₹600 is ✓. A true line-of-best-fit gives ≈₹640, confirming ~₹600 as a fair estimate.

Cells H & I — negative correlation and the zero case

Worked example Example 7 · Two clouds side by side

Set P (hours of TV, exam %): $(1, 90), (2, 80), (3, 70), (4, 60)$ . Set Q (shoe size, exam %): $(6, 72), (9, 68), (7, 74), (11, 70)$ . Question: Classify the correlation in each set.

Forecast: One is a clear downhill; the other is basically noise — which is which?

Set P: as hours rise, score falls in a straight march $90 \to 60$ → negative correlation (strong). Why this step? Downward-trending dots = negative correlation, the mirror image of Example 6.
Set Q: scores stay near 70 no matter the shoe size; the dots form a flat, patternless cloud → no correlation. Why this step? This is the degenerate case — no direction at all. Recognising "no pattern" stops you inventing a fake relationship.
Sanity on causation: even if two things correlate, that does not prove one causes the other — a hidden third variable may drive both. (Shoe size and score share nothing, which is why Q shows nothing.) More in correlation coefficient. Why this step? Guards against the classic "correlation = causation" error.

Verify: In P, $Δ y /Δ x = \frac{60 - 90}{4 - 1} = - 10$ (negative, matches "downhill") ✓. In Q the y-values span only $74 - 68 = 6$ against a wide x-span → near-flat, consistent with "no correlation" ✓.

Cell J — the outlier

Worked example Example 8 · Age vs. reaction time with a rogue point

(age, seconds): $(20, 0.25), (30, 0.28), (40, 0.33), (50, 0.40), (30, 0.60)$ . Question: Identify the outlier and describe the main trend without it.

Forecast: One dot sits way off — which age is repeated with a shockingly slow time?

Plot; find the dot that breaks the pattern. Four dots creep upward; $(30, 0.60)$ floats far above them. Why this step? An outlier is defined as a point far from the overall pattern — the eye finds it fastest on the plot.
Figure 5 — Scatter plot of age (x) vs. reaction time in seconds (y); four teal dots follow a gentle rising dashed line, while an orange "X" marker at (30, 0.60) floats well above the trend as the outlier.
The outlier is $(30, 0.60)$ : another 30-year-old already sits at 0.28 s, so 0.60 s is anomalous. Why this step? Comparing same-x points exposes it: 0.60 vs 0.28 at the same age.
Trend without it: from $(20, 0.25)$ to $(50, 0.40)$ , reaction time gently increases with age. Rate $= \frac{0.40 - 0.25}{50 - 20} = \frac{0.15}{30} = 0.005$ s/year. Why this step? Removing a confirmed error/anomaly reveals the honest trend; possible causes (fatigue, mis-measurement) are then investigated.

Verify: $0.25 + 0.005 \times 30 = 0.40$ = the age-50 value ✓, so the four clean points really are near-linear, isolating $(30, 0.60)$ as the outlier.

Cell K — real-world word problem with prediction

Worked example Example 9 · Predicting from a study-hours line of best fit

A class's data gives the trend line $y = 8 x + 40$ , where $x$ = hours studied and $y$ = test score (%). Question: (a) What does the line predict for 5 hours? (b) What is the meaning of the 8 and the 40?

Forecast: Plug in and interpret — remember what each part of $y = m x + c$ means.

Predict at $x = 5$ : $y = 8 (5) + 40 = 40 + 40 = 80%$ . Why this step? The line of best fit is a machine: feed it x, it returns the expected y. Substitution is the prediction.
Meaning of slope $m = 8$ : each extra hour adds about +8 % on average. Why this step? Slope = $Δ y$ per one unit of $x$ — the rate-of-change idea, now attached to a real sentence.
Meaning of intercept $c = 40$ : at $x = 0$ hours, predicted score is 40 % — the "no study" baseline. Why this step? The intercept $c$ is where the line crosses $x = 0$ ; here it is a meaningful floor.

Verify: At $x = 0$ : $y = 40$ ✓ (matches intercept). At $x = 5$ : $80%$ ✓. Going from 4 to 5 hours: $8 (5) + 40 - (8 (4) + 40) = 8$ = one slope-step ✓.

Cell L — the exam twist (scale trap + extrapolation trap)

Worked example Example 10 · Two traps in one question

A graph shows weekly sales rising from 95 to 97 units over one week, but the y-axis is drawn from 90 to 100, making the line look enormous. A student then argues: "It rose 2 units per week, so in 52 weeks it'll be about 199 units." Question: Expose both mistakes.

Forecast: The picture screams huge, and the future seems obvious — both feelings are wrong.

Scale trap — read the axis numbers, not the slope. The real change is $Δ y = 97 - 95 = 2$ units. Why this step? A zoomed-in axis (90–100) magnifies a tiny rise; only the actual $Δ y$ tells the truth. Always compute the number.
Judge that number fairly in percent. $\frac{2}{95} \times 100 \approx 2.1%$ — a small change, despite the dramatic-looking line. Why this step? Percent puts the change in context relative to the base value, so a "+2" on a base of 95 is correctly seen as minor.
Extrapolation trap — never trust a trend outside the measured range. Assuming the same $+ 2$ /week for 52 weeks predicts $95 + 2 \times 52 = 199$ units. Why this step? We only measured one week. Extrapolation means predicting beyond the data, where the trend usually breaks (sales saturate, stock corrects, plants stop growing). The $199$ figure is unsupported guessing.
State the safe rule. Trust interpolation (predicting inside the data, as in Example 6). Distrust extrapolation (predicting outside it). Here, 52 weeks is far outside one week of data → reject the 199 claim. Why this step? Giving the rule turns the two traps into one reusable habit.

Verify: Actual change $97 - 95 = 2$ ✓ (not "huge"). The naive extrapolation $95 + 2 \times 52 = 199$ ✓ arithmetically — but flagged as unsafe because it leaves the data range.

Cell M — negative x and a vertical line (undefined slope)

Worked example Example 11 · Left of the y-axis, and the divide-by-zero trap

A physics plot uses position on the x-axis (metres, where negative means "left of the origin") against height y (metres). It contains the points $A = (- 4, 1)$ , $B = (- 1, 7)$ , and a wall drawn as the two points $C = (2, 0)$ , $D = (2, 6)$ . Question: (a) Find the slope from $A$ to $B$ . (b) What is the slope of the wall $C \to D$ ?

Forecast: Negative x just means "walk left." But $C$ and $D$ share the same x — what happens to $\frac{Δ y}{Δ x}$ then?

Plot all four points, remembering negative x lands you left of the vertical axis. Why this step? Graphs are not only in the top-right corner; a full coordinate plane has points with negative x. The picture shows $A, B$ on the left, the wall $C, D$ on the right.
Figure 6 — Coordinate plane with drawn x- and y-axes; an orange segment from A(−4,1) to B(−1,7) sits left of the y-axis (slope 2), while a plum vertical segment from C(2,0) to D(2,6) is labelled "delta x = 0, divide by zero!" (undefined slope).
Slope $A \to B$ : $\frac{Δ y}{Δ x} = \frac{7 - 1}{- 1 - ( - 4 )} = \frac{6}{3} = 2$ . Why this step? The subtraction rule is unchanged by negative x: $- 1 - (- 4) = - 1 + 4 = 3$ . A positive slope of 2 means "up 2 for every 1 step right", even though both points sit left of the axis.
Slope $C \to D$ : $\frac{Δ y}{Δ x} = \frac{6 - 0}{2 - 2} = \frac{6}{0}$ → undefined. Why this step? Both points share $x = 2$ , so $Δ x = 0$ . Dividing by zero has no answer — a vertical line has an undefined slope, not zero (zero is the flat case from Example 2). This is the edge case to watch for whenever two data points share the same x.

Verify: $A \to B$ : $- 1 - (- 4) = 3$ and $7 - 1 = 6$ , so $6/3 = 2$ ✓. $C \to D$ : $Δ x = 2 - 2 = 0$ , and division by 0 is undefined → correctly not a number ✓.

Recall Quick self-test

What sign does $Δ y /Δ x$ have for a falling line? ::: Negative. Range of the values 48, 50, 62, 55? ::: $62 - 48 = 14$ . Interpolation vs extrapolation — which is safer and why? ::: Interpolation (predicting inside the data range) is safer; extrapolation goes beyond measured data where trends can break. A flat (constant) line has what rate of change? ::: Exactly zero. A vertical line (two points with the same x) has what slope? ::: Undefined — because $Δ x = 0$ and you cannot divide by zero. Correlation proves causation — true or false? ::: False; a hidden third variable may drive both. What does the ₹ symbol mean on these graphs? ::: Rupees, a unit of money (Indian currency).