Introduction to Graphs — NCERT Solutions, AI Tutor & Practice

The main types of graphs are:

1. Bar graph – uses bars of equal width to compare quantities. Best for comparing amounts (e.g., marks of students in different subjects).
2. Pie chart – a circle divided into sectors. Best for showing parts of a whole (e.g., how a family spends its monthly income).
3. Histogram – bars with no gaps used for continuous class intervals (e.g., number of people in different age groups).
4. Line graph – points joined by line segments. Best for showing how a quantity changes over time (e.g., temperature during a day).

A linear graph is a line graph whose points all lie on a single straight line.

Rise in temperature: $26^\circ\text{C} - 18^\circ\text{C} = 8^\circ\text{C}$

So the temperature rose by $8^\circ$C between 6 a.m. and 10 a.m.

An upward-sloping (rising) line indicates that the quantity (temperature) is increasing as we move from left to right along the time axis. A falling line would mean the temperature is decreasing, and a horizontal line would mean it stays constant.

An ordered pair $(x, y)$ gives the position of a point:

• The first number, $x = 4$, is the x-coordinate (abscissa) — the distance measured along the horizontal x-axis.
• The second number, $y = 3$, is the y-coordinate (ordinate) — the distance measured along the vertical y-axis.

To plot $(4, 3)$:

1. Start at the origin $(0, 0)$.
2. Move $4$ units to the right along the x-axis.
3. From there, move $3$ units up.
4. Mark the point — this is $(4, 3)$.

• $(0, 5)$: The x-coordinate is $0$, so there is no horizontal movement. The point lies on the y-axis, $5$ units above the origin.
• $(7, 0)$: The y-coordinate is $0$, so there is no vertical movement. The point lies on the x-axis, $7$ units to the right of the origin.
• $(0, 0)$: Both coordinates are $0$. This is the origin, where the two axes meet.

Step 1 – Choose axes. Take Year along the x-axis and Amount (₹) along the y-axis.

Step 2 – Plot the points: $(1, 1000)$, $(2, 2000)$, $(3, 3000)$, $(4, 4000)$.

Step 3 – Check the pattern. The increase each year is constant: $2000-1000 = 3000-2000 = 4000-3000 = 1000$

Since the amount rises by ₹1000 for every 1 year, all the points lie on a single straight line. Joining them gives a linear graph.

The relationship is $y = 1000x$, where $x$ is the year and $y$ is the amount.

Using $\text{distance} = \text{speed} \times \text{time} = 60 \times t$:

| Time (h) | 1 | 2 | 3 | 4 | |----------|---|---|---|---| | Distance (km) | 60 | 120 | 180 | 240 |

Plot Time on the x-axis and Distance on the y-axis: $(1,60), (2,120), (3,180), (4,240)$.

Since distance increases by a fixed 60 km each hour, the points lie on a straight line passing through the origin. This is a linear graph and shows that distance is directly proportional to time.

On a distance–time graph, the vertical axis is distance and the horizontal axis is time.

During 2 p.m.–3 p.m. the line is horizontal, meaning the distance stays the same while time keeps increasing.

Since the distance does not change, the cyclist covers no new distance — the cyclist is at rest (stopped) during this hour. A horizontal segment on a distance–time graph always represents a period of rest.

Substitute values into $I = 100t$:

| $t$ (years) | 0 | 1 | 2 | 3 | |-------------|---|---|---|---| | $I$ (₹) | 0 | 100 | 200 | 300 |

The interest increases by ₹100 for every 1-year increase — a constant increase. Therefore the points $(0,0), (1,100), (2,200), (3,300)$ lie on a straight line.

Since the equation is of the form $I = 100t$ (first degree in $t$), the graph is a linear graph passing through the origin.

A scale tells how many units of the actual quantity each division on the graph paper represents.

Why a suitable, uniform scale matters:

1. If the scale is uniform, equal gaps on paper stand for equal changes in value, so the graph reads correctly.
2. If we pick a suitable scale, the whole data fits neatly on the page and is easy to read.

Example: To plot amounts of ₹1000, ₹2000, ₹3000, choosing 1 division = ₹500 is suitable. If instead the first two points used gaps of ₹500 and the next used ₹1000 (non-uniform), a truly linear set of data would appear curved and give wrong conclusions.

Hence a suitable, uniform scale is essential for an accurate graph.

Let $t$ = time in minutes and $y$ = amount of water in litres.

• Starting amount at $t=0$ is $5$ litres.
• Each minute adds $2$ litres, i.e. $2t$ litres in $t$ minutes.

Equation: $y = 5 + 2t$

Water after 6 minutes (put $t = 6$): $y = 5 + 2(6) = 5 + 12 = 17 \text{ litres}$

The graph is a straight line that does not pass through the origin — it crosses the y-axis at $5$ (the initial amount).

Step 1 – Plot the points on graph paper: move right by $x$ and up by $y$ for each pair.

Step 2 – Check the pattern: $\frac{2}{1} = \frac{4}{2} = \frac{6}{3} = \frac{8}{4} = 2$

Each y-coordinate is exactly $2$ times the x-coordinate.

Step 3 – Conclusion: Because the ratio is constant, all four points lie on one straight line through the origin, and the rule is: $y = 2x$

Total hours in a day $= 24$. A full circle $= 360^\circ$.

The central angle for a category is: $\text{Angle} = \frac{\text{hours for that category}}{24} \times 360^\circ$

For Sleep ($8$ hours): $\text{Angle} = \frac{8}{24} \times 360^\circ = \frac{1}{3} \times 360^\circ = 120^\circ$

So the 'Sleep' sector has a central angle of $120^\circ$.