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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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The Cartesian Plane: A graph is drawn on a flat surface called the Cartesian plane, which is divided into four regions by two perpendicular lines. The horizontal line is the xx-axis and the vertical line is the yy-axis.

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The Origin: The point where the xx-axis and yy-axis intersect is called the origin, denoted by OO, with coordinates (0,0)(0, 0). Visually, this is the 'starting point' from which all measurements begin.

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Coordinates of a Point: Every point on a graph is represented by an ordered pair (x,y)(x, y). The first value, xx, is the distance from the yy-axis (abscissa), and the second value, yy, is the distance from the xx-axis (ordinate).

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Line Graph: A line graph displays data that changes continuously over periods of time. It consists of various data points connected by straight line segments, creating a 'jagged' or 'connected' appearance to show trends.

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Linear Graph: A linear graph is a special type of line graph that forms a single, continuous, straight line. This indicates a constant rate of change between the two variables being plotted.

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Plotting Points: To plot a point like (4,3)(4, 3), you start at the origin (0,0)(0, 0), move 44 units to the right along the xx-axis, and then move 33 units vertically upwards. The intersection of these two paths is the location of the point.

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Independent and Dependent Variables: In a linear relationship, the variable that is changed or controlled is the independent variable (plotted on the xx-axis), and the variable that responds is the dependent variable (plotted on the yy-axis). For example, in a 'Distance-Time' graph, time is independent and distance is dependent.

📐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.