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Introduction to Graphs - Line Graphs and Linear Graphs

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A line graph consists of bits of line segments joined consecutively which are used to show continuous data that changes over periods of time. For example, tracking temperature throughout a day.

A line graph showing temperature fluctuations over time.
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A linear graph is a special case of a line graph where the plotted points form a single, straight line. It represents a constant rate of change between two variables, such as distance and time at a constant speed.

A linear graph showing a straight line passing through the origin.
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The location of a point on a graph is given by a pair of coordinates (x,y)(x, y). The xx-coordinate (abscissa) tells the distance from the yy-axis, and the yy-coordinate (ordinate) tells the distance from the xx-axis.

Coordinate plane showing point A at (3,4) with dotted lines to axes.
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Independent variables (the cause) are typically plotted on the horizontal xx-axis, while dependent variables (the effect) are plotted on the vertical yy-axis. For example, when calculating cost based on quantity, quantity is on the xx-axis.

📐Formulae

Coordinates of a point: P(x,y)P(x, y)

Equation of the xx-axis: y=0y = 0

Equation of the yy-axis: x=0x = 0

General form of a linear equation: y=mx+cy = mx + c

Relationship for a square's perimeter: P=4sP = 4s (where ss is the side length)

Simple Interest formula often used in graphs: I=P×R×T100I = \frac{P \times R \times T}{100}

💡Examples

Problem 1:

Plot the points A(2,3)A(2, 3), B(5,3)B(5, 3), and C(5,5)C(5, 5) on a graph. If you join them to form a triangle, find the length of the base ABAB.

Solution:

Step 1: Locate A(2,3)A(2, 3) by moving 22 units right and 33 units up. Step 2: Locate B(5,3)B(5, 3) by moving 55 units right and 33 units up. Step 3: Locate C(5,5)C(5, 5) by moving 55 units right and 55 units up. Step 4: Join AA to BB, BB to CC, and CC to AA. Step 5: Since AA and BB have the same yy-coordinate, the distance ABAB is the difference in their xx-coordinates: 5−2=35 - 2 = 3 units.

Explanation:

This problem demonstrates how to plot points and calculate horizontal distance between points sharing the same yy-level (ordinate).

Problem 2:

A bank gives 10%10\% Simple Interest on deposits. Draw a graph for the interest earned on deposits of ₹100,₹200,₹300₹100, ₹200, ₹300. Use the graph to find the interest on ₹250₹250.

Solution:

Step 1: Calculate interest values: For ₹100,I=10₹100, I = 10; for ₹200,I=20₹200, I = 20; for ₹300,I=30₹300, I = 30. Step 2: Plot points (100,10),(200,20),(300,30)(100, 10), (200, 20), (300, 30) where xx is deposit and yy is interest. Step 3: Join the points with a straight line passing through (0,0)(0, 0). Step 4: To find interest for ₹250₹250, look at x=250x = 250 on the xx-axis, move vertically to the line, then horizontally to the yy-axis. The value is y=25y = 25.

Explanation:

This is a linear graph example where the relationship between deposit and interest is proportional, resulting in a straight line through the origin.

Problem 3:

The following table shows the cost of different quantities of apples. Draw a graph for the data: Quantity (kg): 1,2,3,4,51, 2, 3, 4, 5 Cost (Rs): 20,40,60,80,10020, 40, 60, 80, 100. Is this a linear graph?

Linear graph showing cost vs quantity of apples.

Solution:

  1. Plot the points (1,20),(2,40),(3,60),(4,80),(5,100)(1, 20), (2, 40), (3, 60), (4, 80), (5, 100) on a Cartesian plane.
  2. Connect the points with a line.
  3. Since all the points lie on a single straight line passing through the origin, this is a linear graph.

Explanation:

A linear relationship exists because the cost increases by a fixed amount (Rs 20) for every 1 kg increase in quantity. The equation is y=20xy = 20x.

Problem 4:

Plot the points L(1,1)L(1, 1), M(1,4)M(1, 4), N(4,4)N(4, 4), and O(4,1)O(4, 1) on a graph. Join the points in order. What shape is formed? Find its area.

A square LMNO plotted on a coordinate grid.

Solution:

  1. Plot points L(1,1),M(1,4),N(4,4),O(4,1)L(1,1), M(1,4), N(4,4), O(4,1).
  2. Joining them in order L−M−N−O−LL-M-N-O-L forms a square.
  3. Side length = 4−1=34 - 1 = 3 units.
  4. Area = side×side=3×3=9side \times side = 3 \times 3 = 9 square units.

Explanation:

The coordinates form a closed figure where all sides are equal (33 units) and all angles are 90∘90^{\circ}, confirming it is a square.