NCERT Solutions for Class 8 Maths Chapter 13: Introduction to Graphs — Free PDF

Chapter 13 introduces the Cartesian coordinate system and teaches you how to plot points, read and draw line graphs, and understand linear graphs of the form $y = mx$ or $y = mx + c$.

Graphs are one of the most visual and intuitive tools in mathematics. They help you see patterns, compare data, and understand relationships between variables. This chapter lays the groundwork for coordinate geometry in Classes 9 and 10.

The chapter is divided into two exercises. Exercise 13.1 covers reading and interpreting different types of graphs (bar graphs, pie charts, and line graphs) as well as plotting points on the coordinate plane. Exercise 13.2 focuses on drawing linear graphs from equations like $y = 2x$ or $y = x + 3$. This is one of the more visual and enjoyable chapters in Class 8 Maths.

Cartesian Coordinate System: Named after the mathematician Rene Descartes, this system uses two perpendicular number lines (axes) to locate any point in a plane.

x-axis: The horizontal axis. Values increase to the right and decrease to the left.

y-axis: The vertical axis. Values increase upward and decrease downward.

Origin: The point where the x-axis and y-axis meet, denoted $O(0, 0)$.

Ordered Pair: Every point on the coordinate plane is represented as $(x, y)$, where $x$ is the abscissa (horizontal distance from the origin) and $y$ is the ordinate (vertical distance from the origin). The order matters: $(3, 5)$ and $(5, 3)$ are different points.

Quadrants: The two axes divide the plane into four regions:
- Quadrant I: Both coordinates positive $(+, +)$ — top right.
- Quadrant II: x negative, y positive $(-, +)$ — top left.
- Quadrant III: Both coordinates negative $(-, -)$ — bottom left.
- Quadrant IV: x positive, y negative $(+, -)$ — bottom right.

Points on the axes do not belong to any quadrant. For example, $(5, 0)$ lies on the x-axis, and $(0, -3)$ lies on the y-axis.

Linear Graph: The graph of a linear equation like $y = mx + c$ is always a straight line. The constant $m$ determines the slope (steepness) and $c$ is the y-intercept (where the line crosses the y-axis).

1. Bar Graph: Used for comparing discrete categories. Bars are drawn with equal widths and gaps between them. The height of each bar represents the value. For example, a bar graph might show the number of students in different classes.

2. Pie Chart (Circle Graph): Used to show parts of a whole. The entire circle represents $100\%$ (or $360°$), and each sector represents a fraction of the total. For example, a pie chart might show how a student spends $24$ hours in a day.

3. Line Graph: Used to show data that changes over time (continuous data). Points are plotted and connected with straight line segments. For example, a city's temperature recorded every hour over a day. Line graphs are excellent for spotting trends — rising, falling, or staying constant.

4. Linear Graph: A specific type of graph representing a linear equation. Unlike a general line graph (which may zig-zag), a linear graph is always a perfectly straight line. Every point on this line satisfies the equation.

The key distinction to remember: a line graph connects data points (which may or may not form a straight line), while a linear graph is the graph of a specific equation and is always perfectly straight.

**Q1. Plot the following points on a graph: $A(2, 3)$, $B(-1, 4)$, $C(-3, -2)$, $D(4, -1)$.**

Solution:

• $A(2, 3)$: Move $2$ units right on x-axis and $3$ units up on y-axis. This point lies in Quadrant I $(+, +)$.
  - $B(-1, 4)$: Move $1$ unit left and $4$ units up. This point lies in Quadrant II $(-, +)$.
  - $C(-3, -2)$: Move $3$ units left and $2$ units down. This point lies in Quadrant III $(-, -)$.
  - $D(4, -1)$: Move $4$ units right and $1$ unit down. This point lies in Quadrant IV $(+, -)$.

Each point falls in a different quadrant, which makes this a good exercise for understanding the sign conventions.

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**Q2. From the line graph showing a city's temperature over $5$ days, the temperatures were: Mon $25°$C, Tue $28°$C, Wed $30°$C, Thu $27°$C, Fri $26°$C. On which day was the temperature highest? Between which consecutive days was the rise in temperature the greatest?**

Solution:

The temperature was highest on Wednesday at $30°$C. This is the highest point on the line graph.

The rise between consecutive days:
- Mon to Tue: $28 - 25 = 3°$C
- Tue to Wed: $30 - 28 = 2°$C
- Wed to Thu: temperature fell by $3°$C
- Thu to Fri: temperature fell by $1°$C

The greatest rise was $3°$C, from Monday to Tuesday.

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**Q3. State the quadrant for each: $(3, -5)$, $(-2, -7)$, $(-4, 3)$, $(6, 1)$.**

Solution:

• $(3, -5)$: $x$ positive, $y$ negative $\rightarrow$ Quadrant IV
  - $(-2, -7)$: both negative $\rightarrow$ Quadrant III
  - $(-4, 3)$: $x$ negative, $y$ positive $\rightarrow$ Quadrant II
  - $(6, 1)$: both positive $\rightarrow$ Quadrant I

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**Q4. Write the coordinates of a point that lies (a) on the x-axis at $x = 5$, (b) on the y-axis at $y = -3$.**

Solution:

(a) Points on the x-axis have $y = 0$. So the point is $(5, 0)$.
(b) Points on the y-axis have $x = 0$. So the point is $(0, -3)$.

**Q1. Draw the graph of $y = 2x$.**

Solution:

Step 1: Make a table of values by choosing convenient values of $x$:

Step 2: Plot points $(-2, -4)$, $(-1, -2)$, $(0, 0)$, $(1, 2)$, $(3, 6)$.

Step 3: Join them with a straight line using a ruler.

The graph is a straight line passing through the origin. Since the equation is $y = 2x$ (no constant term), the line always passes through $(0, 0)$.

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**Q2. Draw the graph of $y = x + 3$.**

Solution:

Plot points $(-3, 0)$, $(-1, 2)$, $(0, 3)$, $(2, 5)$, $(4, 7)$ and join them with a straight line.

The line crosses the y-axis at $(0, 3)$ — this is the y-intercept. It crosses the x-axis at $(-3, 0)$ — this is the x-intercept.

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**Q3. Does the point $(2, 5)$ lie on the graph of $y = 2x + 1$?**

Solution:

Substitute $x = 2$ into the equation: $y = 2(2) + 1 = 5$.

Since we get $y = 5$, which matches the given point, yes — $(2, 5)$ does lie on the graph of $y = 2x + 1$.

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**Q4. Draw the graph of $y = -x$.**

Solution:

Plot and join. The line passes through the origin and slopes downward from left to right. This shows that when the coefficient of $x$ is negative, the line slopes downward.

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**Q5. From the graph of $y = 3x - 2$, find the value of $y$ when $x = 4$, and the value of $x$ when $y = 7$.**

Solution:

When $x = 4$: $y = 3(4) - 2 = 10$.

When $y = 7$: $7 = 3x - 2 \Rightarrow 9 = 3x \Rightarrow x = 3$.

On the graph, draw a vertical line from $x = 4$ to find $y = 10$, and a horizontal line from $y = 7$ to find $x = 3$.

**Example 1. Plot the following points and identify the shape: $A(1, 1)$, $B(1, 4)$, $C(4, 4)$, $D(4, 1)$.**

Solution:

Plotting these four points and joining them in order gives a square with side length $3$ units. Each side is either horizontal or vertical, making it easy to see the square shape on the coordinate plane.

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**Example 2. The graph of $y = mx$ passes through $(3, 9)$. Find $m$.**

Solution:

Substitute the point into the equation: $9 = m \times 3$, so $m = 3$.

The equation is $y = 3x$.

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**Example 3. A shopkeeper earns a profit of Rs $y$ on selling $x$ items, given by $y = 5x - 20$. (a) What is the profit on $10$ items? (b) How many items must be sold to break even (zero profit)?**

Solution:

(a) $y = 5(10) - 20 = 50 - 20 = 30$. Profit $= \text{Rs } 30$.

(b) Break even means $y = 0$: $0 = 5x - 20 \Rightarrow x = 4$. The shopkeeper must sell at least $4$ items to break even.

On the graph, the break-even point is where the line crosses the x-axis at $(4, 0)$.

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**Example 4. Which of the following points lie on the line $y = 2x - 3$? (a) $(1, -1)$ (b) $(2, 2)$ (c) $(3, 3)$ (d) $(0, -3)$.**

Solution:

Substitute each point:
- (a) $y = 2(1) - 3 = -1$. Point gives $y = -1$. Lies on the line.
- (b) $y = 2(2) - 3 = 1$. Point gives $y = 2$. Does not lie on the line.
- (c) $y = 2(3) - 3 = 3$. Point gives $y = 3$. Lies on the line.
- (d) $y = 2(0) - 3 = -3$. Point gives $y = -3$. Lies on the line.

Mistake 1: Swapping x and y coordinates.
The point $(3, 5)$ means $x = 3, y = 5$ — move $3$ right and $5$ up. Writing $(5, 3)$ instead gives a completely different point. Always remember: x comes first (horizontal), y comes second (vertical).

Mistake 2: Forgetting to label axes and title.
In exams, marks are deducted for unlabelled graphs. Always write the variable name on each axis (e.g., "Time (hours)" on x-axis, "Distance (km)" on y-axis) and give the graph a title.

Mistake 3: Plotting too few points for a linear graph.
While two points determine a line, always plot at least three points. The third point serves as a verification — if it does not fall on the same line as the first two, you have made a calculation error.

Mistake 4: Confusing line graphs with linear graphs.
A line graph connects data points and may zig-zag. A linear graph represents an equation and is always a perfectly straight line. They are not the same thing.

Mistake 5: Not choosing appropriate scale.
If your values range from $0$ to $100$, do not use a scale of $1$ unit per square — the graph will be too large. Choose a scale that fits the graph paper (e.g., $1$ square $= 10$ units).

Mistake 6: Errors in sign when plotting negative coordinates.
$(-3, 2)$ means $3$ units to the LEFT and $2$ units UP. Students sometimes go right for a negative x-coordinate. Pay close attention to the signs.

Quadrant sign convention:

Points on axes:
- Any point on the x-axis has $y = 0$: $(a, 0)$
- Any point on the y-axis has $x = 0$: $(0, b)$
- The origin has both coordinates zero: $(0, 0)$

**Key features of a linear graph $y = mx + c$:**
- $m$ (slope): positive $\Rightarrow$ line goes upward; negative $\Rightarrow$ line goes downward; zero $\Rightarrow$ horizontal line
- $c$ (y-intercept): the point $(0, c)$ where the line crosses the y-axis
- If $c = 0$ ($y = mx$): line passes through the origin
- x-intercept: set $y = 0$ and solve for $x$: $x = -c/m$

How to quickly plot a linear graph:
1. Find the y-intercept by setting $x = 0$: gives point $(0, c)$.
2. Find the x-intercept by setting $y = 0$: gives point $(-c/m, 0)$.
3. Plot a third point for verification.
4. Join with a ruler.

This method of using intercepts is faster than making a full table of values.

Q1. Plot the points $P(0, 4)$, $Q(-3, 0)$, $R(2, -5)$. State which quadrant or axis each point lies on.

Answer: $P(0, 4)$ lies on the y-axis. $Q(-3, 0)$ lies on the x-axis. $R(2, -5)$ lies in Quadrant IV.

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Q2. Draw the graph of $y = x - 2$. Where does the line cross the x-axis and y-axis?

Answer: Table: $x = 0 \Rightarrow y = -2$; $x = 2 \Rightarrow y = 0$; $x = 4 \Rightarrow y = 2$. The line crosses the y-axis at $(0, -2)$ and the x-axis at $(2, 0)$.

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Q3. Does the point $(4, 7)$ lie on the graph of $y = 2x - 1$?

Answer: Substitute: $y = 2(4) - 1 = 7$. Yes, it does.

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Q4. The graph of $y = mx$ passes through the point $(2, 6)$. Find $m$ and then find $y$ when $x = 5$.

Answer: $6 = 2m \Rightarrow m = 3$. When $x = 5$: $y = 3 \times 5 = 15$.

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Q5. A taxi charges Rs $y$ for a ride of $x$ km, where $y = 10x + 50$. (a) What is the minimum charge? (b) What is the charge for $8$ km?

Answer: (a) Minimum charge (at $x = 0$) $= \text{Rs } 50$ (the base fare). (b) $y = 10(8) + 50 = \text{Rs } 130$.

• The Cartesian plane has four quadrants, determined by the signs of $x$ and $y$ coordinates.
  - In an ordered pair $(x, y)$, the x-coordinate (abscissa) always comes first.
  - Line graphs show trends over time; linear graphs represent equations like $y = mx + c$.
  - To plot a linear graph: make a table, plot at least $3$ points, join with a straight line.
  - To check if a point lies on a line, substitute its coordinates into the equation.
  - Points on the x-axis have $y = 0$; points on the y-axis have $x = 0$.
  - Always label axes, choose an appropriate scale, and use a ruler for straight lines.

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