The Steepest Part of a Line Tells the Real Story

The steepest part. A line graph's slope shows how fast a quantity is changing, so the steepest stretch marks the interval where the biggest change happened — often the one thing the graph was drawn to reveal.

The points give values; the slopes between them give the story. This page covers everything in the CBSE Class 8 Mathematics chapter's second part: drawing a line graph, reading it, comparing two series, and interpreting an infographic.

Put time on the horizontal axis and the changing quantity on the vertical axis, plot each point, then join consecutive points with straight line segments.

Choosing the scale is the step that needs care. Divide the largest value by the space available on that axis.

Worked example. Temperature recorded at 6 am on five days: $18°C$, $20°C$, $23°C$, $22°C$, $26°C$.

The largest value is $26$. With about $7$ cm of vertical space:

$$\frac{26}{7}\approx 3.7,\text{so take }1\text{ cm}=4°C$$

Then $26°C$ is drawn at $\frac{26}{4}=6.5$ cm, which fits comfortably. A rounder choice of $1$ cm $=5°C$ puts it at $5.2$ cm, also fine.

Four things must appear on the finished graph, and each is marked:

- Both axes labelled with the quantity and its unit — Day and Temperature in °C
- The scale stated in words, as 1 cm = 4°C on the vertical axis
- Points plotted accurately and joined by straight segments
- A title naming what the graph shows

Work out the scale before plotting anything. A scale chosen afterwards usually leaves the tallest point off the page or the whole graph squashed into two centimetres, and either way it has to be redrawn — whereas one division at the start settles it.

For a value, go up from the time on the horizontal axis to the line, then across to the vertical axis and read against the scale.

For the greatest change, compare the differences between consecutive readings — not the heights of the points.

Worked example, using the five temperatures $18,20,23,22,26$:

- Day 1 to 2: $20-18=+2$
- Day 2 to 3: $23-20=+3$
- Day 3 to 4: $22-23=-1$
- Day 4 to 5: $26-22=+4$

The greatest increase is from day 4 to day 5, a rise of $4°C$, and that stretch is the steepest on the graph. The only decrease is from day 3 to day 4.

The overall trend is upward, from $18°C$ to $26°C$ — a total rise of $8°C$ across the five days.

Reading between points. For a quantity that changes smoothly, a point halfway along the segment from $18°C$ to $20°C$ suggests about $19°C$ at the midpoint. That estimate is reasonable for temperature, which varies continuously.

A rising line means increase, a falling line means decrease, and a flat line means no change at all.

The distinction that decides these questions is between the highest point and the biggest change. Day 5 has the highest temperature, but the biggest change is the jump into day 5 — so a question asking where the quantity rose fastest wants the interval, not the peak.

Plot both with the same scale on the same axes, draw each in a different style, and supply a key naming which line is which.

Worked example. Attendance in two classes over five days:

- Class A: $38,40,36,42,44$
- Class B: $42,41,39,38,36$

Reading the two lines together:

- Class B started higher, at $42$ against $38$
- The lines cross between day 2 and day 3, after which A is above B
- Class A's trend is upward, from $38$ to $44$
- Class B's trend is downward, from $42$ to $36$
- The largest gap is on day 5: $44-36=8$ students

A supported conclusion names the numbers: Class A's attendance improved from 38 to 44 while Class B's fell from 42 to 36, so by day 5 Class A was ahead by 8 students.

The crossing point is usually the most interesting feature, because it marks where the comparison reverses.

Two requirements are marked every time. A key is essential — two lines with no key cannot be read at all. And both series must use the same scale, or the comparison is meaningless: drawing Class B against a stretched vertical axis would make its decline look dramatic when it is a fall of six students.

An infographic presents data using pictures, charts and figures together. Read it in a fixed order, then look for what it leaves out.

Reading it:

1. Read the title to find what it is about
2. Identify each chart or picture and what it measures
3. Check the units and scale on every axis or symbol
4. Note the source and what period the data covers
5. Read off the specific figures the question asks for

Formulating a further question means noticing what the data cannot tell you. Good further questions:

- Does this pattern hold for other regions, or only the one shown?
- What happened in the years between the ones displayed?
- How many people were surveyed, and were they chosen fairly?
- Is the change shown large compared with the normal variation?
- Does the vertical axis start at zero?

That last question is the sharpest one, and it is worth asking of any chart. An axis that begins part-way up exaggerates small differences: two bars of $48$ and $50$ look nearly equal from zero, but dramatically different if the axis starts at $45$. Every value is plotted honestly and the picture still misleads.

So a good further question is not a complaint but a test of what the data supports. The graph above shows attendance for five days in two classes — it cannot tell you why the trends differ, whether they continued, or whether other classes behaved the same way, and naming one of those gaps is exactly the kind of question the syllabus is asking for.