Introduction to Graphs

Pick up any newspaper's business page, a weather app, or a doctor's case sheet, and you will find the same trick used to make numbers speak faster: a graph. A jagged line rising and falling tells the story of a fever, a share price, or a cricket innings in one glance, when the same data as a table would take a full minute to read. This chapter is about learning to read that trick, and then to draw it yourself. We start with the simplest and commonest picture, the line graph, where points recorded over time are joined by straight segments — a doctor's temperature chart, a company's yearly sales, the runs scored by two batsmen match after match. Reading such a graph well means being able to say exactly what happened between 10 a.m. and 2 p.m., or when two quantities were equal, purely by tracing lines to the two axes. The chapter then asks a sharper question: when do two changing quantities — litres of petrol bought and money paid, or a sum deposited and the interest it earns — vary together so simply that their graph is a perfectly straight line passing through the very corner where both quantities are zero? That pattern is called direct variation, and it lets a shopkeeper or a bank predict an unknown value just by drawing one straight line and reading off a point on it. By the end, you will be able to plot related quantities as points fixed by an x-coordinate and a y-coordinate, choose a sensible scale, tell a general line graph apart from a strictly linear one, and test — using nothing more than a ratio — whether a relationship is direct variation. These are the foundation for every later coordinate-geometry topic.

Key takeaways

A line graph displays data recorded over time by plotting points and joining consecutive points with straight segments; the standard way to show a trend.
Every point on a graph is fixed by an x-coordinate (horizontal axis) and a y-coordinate (vertical axis), always written in that order as $(x, y)$ .
The horizontal x-axis and vertical y-axis cross at the origin, $(0, 0)$ .
The quantity we choose values for is the independent variable (x-axis); the one that responds is the dependent variable (y-axis).
A linear graph is a line graph that turns out to be one single unbroken straight line; not every line graph is straight.
Two quantities are in direct variation exactly when $y = kx$ for a fixed constant $k$ ; their graph is then a straight line through the origin, and $y/x$ equals $k$ for every pair.
On a distance-time graph, a horizontal (flat) segment means the distance is not changing while time passes — rest, not slow motion.
A suitable, evenly spaced scale on each axis — not necessarily the same on both — is what makes a graph readable and accurate.
A straight-line graph that does not pass through the origin is still a linear graph, but it is not direct variation.
Key idea: every related pair of quantities can be plotted as points fixed by an x-coordinate and a y-coordinate, and when the two quantities obey y = kx their graph is a straight line through the origin — the signature of direct variation.

The core insight

Any two related quantities can be pictured as points $(x, y)$ on a graph, and the special, highly useful case where $y = kx$ for a constant $k$ produces a straight line through the origin, the graphical fingerprint of direct variation.

What this chapter covers

This chapter builds up graph skills in four connected stages.

Line graphs and how to read them — plotting recorded data as points joined by segments, and extracting facts (highest value, when two quantities were equal, rising or falling trend) directly from the picture.
Independent and dependent variables — deciding which quantity is freely chosen (x-axis) and which one responds (y-axis), in everyday situations like a petrol purchase.
Distance-time graphs — where a horizontal segment has a precise meaning (no motion), and the shape tells you when speed was constant, increasing, or zero.
Linear graphs and direct variation — testing with a ratio check, and using straight-line-through-the-origin graphs of quantities in direct proportion, such as cost against quantity bought.

Two worked derivations turn NCERT's plotting steps into a justified method: how coordinates fix a point uniquely, and why direct variation must graph as a straight line through the origin.

The core ideas

What a graph is, and why we use one

A graph is a visual picture of numerical data, drawn so a trend or comparison can be read off in a glance rather than worked out from a table. The most familiar type here is the line graph: each piece of recorded data becomes a point on a squared grid, and consecutive points are joined by straight segments. A doctor's four-hourly temperature record, plotted this way, becomes a "time-temperature graph" that instantly shows when a fever was rising and falling, even though no single number in the table says so. Every point on such a graph needs exactly two pieces of information to fix its position: how far along the horizontal direction, and how far up the vertical direction. These two numbers are written together, in a fixed order, as an ordered pair.

P = (x, y)

where: $x$ , called the x-coordinate, is the point's distance measured along the horizontal axis (the x-axis), and $y$ , called the y-coordinate, is its distance measured along the vertical axis (the y-axis). The order matters — $(x, y)$ and $(y, x)$ are different points in general — and the point where both coordinates are zero, $(0, 0)$ , is called the origin.

Independent and dependent variables

Many everyday situations involve one quantity that changes because another one does. If you buy more units of electricity, your bill goes up. The quantity of electricity used is under your control, so it is the independent variable, and the bill, which responds, is the dependent variable. By rule, the independent variable is marked along the x-axis and the dependent variable along the y-axis. In "litres of petrol bought" versus "amount paid," you choose the litres, and the cost follows, so litres is independent and cost is dependent.

Reading a line graph: peaks, valleys, and steady trends

Once a line graph is drawn, most questions are answered by tracing along one axis to the graph, then across to the other. A "peak" shows a maximum value at that moment; a "valley" shows a minimum. Two lines that cross or touch at a point mean the two quantities being compared were equal at that x-value. A line with many sharp ups and downs describes an inconsistent quantity, while one that stays close to a steady level describes a more consistent one — the comparison used when judging which of two batsmen scores more reliably match after match.

Distance-time graphs: reading motion from a picture

A common and useful line graph plots distance travelled (y-axis) against time elapsed (x-axis). Its shape carries specific meaning: a segment sloping upward means distance is increasing, so the object is moving; a steeper segment covers more distance in the same time, a faster speed; and — the detail students most often get wrong — a perfectly horizontal segment means the distance is not changing at all while time passes, which can only mean the object has stopped completely, not that it is moving slowly.

Figure

A distance time graph for a car travelling from city P to city Q showing distance from P on the vertical axis against time on the horizontal axis, rising steeply, then rising more gently, then a flat rest segment, then rising again to reach 350 kilometres

Linear graphs and direct variation

Sometimes the points of a graph, once plotted, all fall on one single, unbroken straight line. Such a graph is a linear graph. Every linear graph is a line graph, but not every line graph is linear — the temperature or runs graphs above are not straight, so they are line graphs without being linear. An especially important case arises when two quantities $x$ and $y$ change together in a fixed proportion — double one and the other doubles too. This relationship is written algebraically as

y = kx

where: $k$ is a fixed number called the constant of variation (the price per litre of petrol, say, or the rate of simple interest per rupee deposited), and $x, y$ range over the corresponding pairs of values. Quantities satisfying this equation are in direct variation, and — as the second derivation below proves — their graph is always a straight line through the origin $(0,0)$ .

Figure

A direct variation graph of cost of petrol against number of litres bought, a straight line through the origin passing through the points ten litres five hundred rupees fifteen litres seven hundred fifty rupees twenty litres one thousand rupees and twenty five litres one thousand two hundred fifty rupees, with a dashed reading for twelve litres giving six hundred rupees

Because the line passes through the origin, the point $(0, 0)$ itself has a meaning: zero litres of petrol cost nothing, zero rupees deposited earns zero interest. That single geometric fact — passing through the origin — is exactly what separates direct variation from an ordinary straight-line (linear) graph, and it is proved formally below.

Derivation ALocating a point using its x-coordinate and y-coordinate

The goal is to show, using nothing beyond measuring along two perpendicular reference lines, that every point on graph paper corresponds to exactly one ordered pair $(x, y)$ , and that any ordered pair fixes exactly one point. This correspondence is the quiet fact underneath every graph in this chapter — every instruction of the form "plot the point $(10, 500)$ " depends on it.

Step 1. Draw the two reference lines: a horizontal x-axis and a vertical y-axis, meeting at the origin, $(0, 0)$ . Mark a suitable, evenly spaced scale on each axis — equal steps must represent equal amounts, though the two axes need not share the same scale, since one might measure hours and the other hundreds of rupees.

Step 2. To plot $(10, 500)$ , start at the origin and move along the x-axis by the first number, $10$ units on the chosen scale. This fixes a vertical line through that mark.

Step 3. From that mark, move parallel to the y-axis by the second number, $500$ units on the vertical scale. This fixes a horizontal line through the new position. The single point where the two lines meet is $(10, 500)$ — the only point lying on both, so the construction gives one unique point.

Step 4. The reverse process reads a point's coordinates: from a marked point, a line parallel to the y-axis meets the x-axis at the x-coordinate, and a line parallel to the x-axis meets the y-axis at the y-coordinate. Since two distinct lines meet in at most one point, this pair of readings is uniquely fixed by the point — completing the two-way correspondence between points and ordered pairs.

Step 5. Note the order convention: $(x, y)$ always lists the horizontal reading first, the vertical reading second; swapping them marks a different point. A table of values must always be read as (independent-variable value, dependent-variable value) before it is plotted.

Go deeper · Derivation B · 1 minWhy a direct-variation graph is a straight line through the origin

Derivation BWhy a direct-variation graph is a straight line through the origin

The goal is to prove a fact every worked example here relies on: if $x$ and $y$ satisfy $y = kx$ for a fixed constant $k$ , plotting all their corresponding pairs gives points lying on one straight line through the origin. This turns NCERT's own question — "why does this graph pass through the origin?" — into a short argument using ratios, already familiar from direct proportion.

Step 1. Check the origin lies on the graph. Substituting $x = 0$ into $y = kx$ gives $y = 0$ . So $(0, 0)$ always satisfies the relation, matching the everyday reading that zero litres of petrol cost zero rupees.

Step 2. Show the ratio $y/x$ is the same fixed number $k$ for every other pair. Take two pairs $(x_1, y_1)$ and $(x_2, y_2)$ satisfying $y = kx$ , with $x_1, x_2 \neq 0$ . Then $y_1 = kx_1$ and $y_2 = kx_2$ , so dividing each by its own $x$ -value gives $\dfrac{y_1}{x_1} = \dfrac{y_2}{x_2} = k$ . This is exactly the constant-ratio test used on the petrol table: $\frac{500}{10} = \frac{750}{15} = \frac{1000}{20} = \frac{1250}{25} = 50$ .

Step 3. Turn the equal-ratio fact into "lying on one straight line." Consider the origin $O$ and any two points $A = (x_1, y_1)$ , $B = (x_2, y_2)$ satisfying $y = kx$ . Dropping a perpendicular from $A$ and from $B$ to the x-axis gives two right triangles sharing the angle at $O$ , with the same ratio of vertical to horizontal side, $k$ , in both — so the triangles are similar, and $OA$ , $OB$ make the same angle with the x-axis. Hence $O$ , $A$ , $B$ lie on one line, and the same argument holds for any such pair.

Step 4. The converse holds too: if a graph is a straight line through the origin, then any point $(x, y)$ on it besides the origin gives $y/x$ equal to the same fixed $k$ , precisely $y = kx$ . So "straight line through the origin" and "direct variation" describe exactly the same graphs — checking either test is equally valid.

Worked examples

Board level - reading a distance-time graph for a journey from City P to City Q

The graph above shows a car's distance from City P at different times, travelling to City Q, 350 km away: $0$ km at 8 a.m., $50$ km at 9 a.m., $150$ km at 10 a.m., $200$ km at 11 a.m., still $200$ km at 12 noon, and $350$ km at 2 p.m. Find (a) the distance in the first hour, (b) the distance in the second hour, (c) whether the car stopped, and (d) when it reached City Q.

Step 1. Read the first hour, 8 a.m. to 9 a.m.: distance goes from $0$ km to $50$ km, so the car covered $50 - 0 = 50$ km.

Step 2. Read the second hour, 9 a.m. to 10 a.m.: distance goes from $50$ km to $150$ km, a change of $100$ km — faster than the first hour, since the graph is steeper there.

Step 3. Compare 11 a.m. and 12 noon: distance reads $200$ km at both times, unchanged while an hour passes. A flat segment means no motion, so the car was stationary for that hour.

Step 4. Locate where the graph reaches $350$ km, the full distance to City Q, and read the time under that point. Answer: the car covered 50 km in the first hour, 100 km in the second hour, it rested for one hour between 11 a.m. and 12 noon, and it reached City Q at 2 p.m.

Board level - using a direct-variation graph to read unknown values

A bank pays 10% simple interest a year on senior-citizen deposits: Rs 100 earns Rs 10, Rs 200 earns Rs 20, Rs 300 earns Rs 30, Rs 500 earns Rs 50. Using direct variation, find (a) the annual interest on a deposit of Rs 250, and (b) the deposit needed to earn Rs 70 annual interest.

Step 1. Confirm direct variation using the ratio test: $\frac{10}{100} = \frac{20}{200} = \frac{30}{300} = \frac{50}{500} = 0.1$ . The constant ratio $k = 0.1$ means interest $I$ and deposit $D$ satisfy $I = 0.1 \times D$ , a straight line through the origin.

Step 2. Find the interest on Rs 250: $I = 0.1 \times 250 = 25$ . On the graph this is the point above $D = 250$ , read across to the vertical axis.

Step 3. Find the deposit for Rs 70 interest by solving $70 = 0.1 \times D$ , giving $D = 700$ . Answer: a deposit of Rs 250 earns Rs 25 annual interest, and Rs 700 must be deposited to earn Rs 70 annual interest.

Higher-order thinking (HOTS / olympiad-style) - is the side-area graph of a square also a direct variation

For a square of side $s$ cm, perimeter $P = 4s$ and area $A = s^2$ . For $s = 2, 3, 4, 5, 6$ : perimeters $8, 12, 16, 20, 24$ ; areas $4, 9, 16, 25, 36$ . Without plotting, determine whether perimeter-against-side and area-against-side are (i) linear graphs, and (ii) direct variations.

Step 1. Test the perimeter-side data with the ratio test: $\frac{8}{2} = \frac{12}{3} = \frac{16}{4} = \frac{20}{5} = \frac{24}{6} = 4$ . Every ratio equals the same constant $k = 4$ , matching $P = 4s$ . Equal ratios throughout means all the points lie on one straight line through the origin, so the perimeter-side graph is both a linear graph and a direct variation.

Step 2. Test the area-side data the same way: $\frac{4}{2} = 2$ , $\frac{9}{3} = 3$ , $\frac{16}{4} = 4$ , $\frac{25}{5} = 5$ , $\frac{36}{6} = 6$ . The ratio keeps changing, not a fixed number, so the points cannot all lie on one straight line through the origin — this graph is neither linear nor a direct variation.

Step 3. Explain algebraically, without plotting: $A = s^2$ is not of the form $y = kx$ for any fixed $k$ , since the right side is $s$ multiplied by itself, not by a constant — the "rate" $A/s = s$ itself keeps changing. Such a squared relation always produces a curve, never a straight line. Answer: the perimeter-side graph is a linear graph and a direct variation (k = 4); the area-side graph is neither linear nor a direct variation, because A/s is not constant. Timing note: about 3 minutes, without drawing anything. The ratio test is the fast, exam-safe way to settle "is this graph linear or a direct variation" — compute $y/x$ for every pair; the same number every time means yes to both, a changing ratio means no, and the algebraic form (a square, a fraction, an added constant) confirms whether it is even a straight line at all.

Common traps

Students lose marks here less from calculation errors and more from conceptual mix-ups.

Reversing the coordinates. Writing $(500, 10)$ when the deposit is $100$ and interest is $10$ mixes up which axis a number belongs to; always write (independent-variable value, dependent-variable value).
Uneven or careless scale. Marking an axis so equal grid steps do not represent equal amounts distorts every reading taken off the graph afterwards.
Treating "line graph" and "linear graph" as the same word. A line graph only needs points joined by segments; it becomes linear only if every point lies on one straight line. A temperature chart is a line graph but almost never a linear one.
Calling any straight line a direct variation. A straight line not passing through the origin (say, one with a fixed starting charge) is still a linear graph, but not direct variation, since $y = kx$ forces the graph through $(0,0)$ .
Misreading a flat segment as "slow motion." A horizontal segment means distance is not changing at all — rest, not reduced speed; a small nonzero rise represents slow motion.
Extrapolating far beyond the given data. Reading a graph confidently for values well outside the plotted range assumes the pattern continues unchanged, which is not always safe.

You should be able to…

read numerical facts directly off a line graph, such as maximum and minimum values, and where two lines meet
decide, for a real situation, which quantity is independent and which is dependent
plot a table of paired values as points $(x, y)$ using a suitable, consistent scale on each axis
explain the difference between a general line graph and a linear (straight-line) graph
test whether two quantities are in direct variation by checking that $y/x$ is constant
state, and use, the fact that a direct-variation graph is a straight line through the origin
interpret a horizontal segment on a distance-time graph correctly as a period of no motion
use a linear or direct-variation graph to read off an unknown value not directly given in the table

Ask the tutor: Try plotting your own two related quantities from daily life — like days of the week against your study hours — and ask the tutor to check whether your graph is a linear graph, and whether it shows direct variation.

What exactly is the difference between a line graph and a linear graph?

A line graph is any graph made by plotting data points and joining consecutive ones with straight segments — a method of drawing, and the result can zig-zag, like a fever chart. A linear graph is a stronger claim: every point must land on one unbroken straight line, with no bends anywhere. So every linear graph is a line graph, but a line graph is only "linear" if it turns out perfectly straight. When a question asks "is this a linear graph," it wants you to check whether the points genuinely line up, not just whether segments were drawn between them.

Why does a direct-variation graph always pass through the origin?

Because direct variation means $y = kx$ for a fixed number $k$ , and putting $x = 0$ into that equation always gives $y = 0$ . So $(0, 0)$ is always one of the valid pairs whenever the relationship holds — zero of the independent quantity must go with zero of the dependent quantity, like zero litres of petrol costing zero rupees. This is also why a straight line that does not start at the origin (say, a taxi fare with a fixed base charge) cannot be a direct variation, even though it is perfectly straight: it fails the $x=0 \Rightarrow y=0$ test.

How do I choose a good scale when I have to draw the graph myself?

Look at the largest value you need to fit on each axis, then pick a round number of units per grid square (1, 2, 5, 10, 50, and so on) so the largest value lands comfortably within the sheet without cramming everything into a corner or running off the page. The two axes almost never need the same scale as each other — time in hours and money in hundreds of rupees are different kinds of quantity — but the scale on any one axis must stay the same all the way along it, or every reading taken from the graph becomes unreliable.

Can the independent variable ever be plotted on the y-axis instead of the x-axis?

By the convention used throughout this chapter, no — the independent variable (the one you choose or control) goes on the horizontal x-axis, and the dependent variable (the one that responds) goes on the vertical y-axis. Following this consistently lets anyone reading the graph immediately see which quantity is "driving" the other, without a caption. It is a convention, not a mathematical necessity, but exam answers are expected to follow it, and swapping the axes without saying so is treated as an error.