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Introduction to Graphs - Reading and Interpreting Graphs

Grade 8CBSE

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

🔑Concepts

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The Cartesian Plane consists of two perpendicular number lines: the horizontal line called the xx-axis and the vertical line called the yy-axis. Their point of intersection is called the Origin, denoted by O(0,0)O(0, 0). Visually, this divides the plane into four quadrants.

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Coordinates of a point are written as an ordered pair (x,y)(x, y). The first number xx (abscissa) represents the horizontal distance from the yy-axis, and the second number yy (ordinate) represents the vertical distance from the xx-axis. For example, to plot (3,5)(3, 5), you move 33 units right and 55 units up.

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A Line Graph is used to show how a dependent variable changes with respect to an independent variable over a period of time. It consists of data points connected by straight line segments. Visually, a continuous straight line indicates a linear relationship.

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Independent and Dependent Variables: The independent variable (e.g., time, which changes on its own) is usually plotted on the horizontal xx-axis, while the dependent variable (e.g., distance, which depends on time) is plotted on the vertical yy-axis.

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Reading Trends: The slope or direction of the line provides information about the data. A line rising from left to right indicates an increase, a line falling indicates a decrease, and a horizontal line indicates that the yy-value remains constant despite changes in xx.

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Choosing a Scale: Since graph paper has limited space, we use a scale (e.g., 1 cm=10 units1 \text{ cm} = 10 \text{ units}) to represent large numbers. A 'kink' or a 'zigzag' line on an axis indicates that the values between zero and the first marked value are being skipped.

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Linear Graphs: If all the plotted points of a data set lie on a single straight line, the graph is called a linear graph. This happens when the change in yy is proportional to the change in xx.

📐Formulae

Coordinate Point Representation: P(x,y)P(x, y)

Origin Coordinates: O(0,0)O(0, 0)

General Equation of a Linear Graph: y=mx+cy = mx + c

Slope (Rate of Change): m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Points on the xx-axis: (x,0)(x, 0)

Points on the yy-axis: (0,y)(0, y)

💡Examples

Problem 1:

A bank gives 10%10\% Simple Interest on deposits. Draw a graph for the relationship between the sum deposited and the interest earned. From the graph, find the interest on a deposit of Rs.250Rs. 250.

Solution:

  1. Let Deposit be xx and Interest be yy. The relation is y=10100×x=0.1xy = \frac{10}{100} \times x = 0.1x.
  2. Create points: If x=100,y=10x=100, y=10; if x=200,y=20x=200, y=20; if x=300,y=30x=300, y=30.
  3. Plot points (100,10),(200,20),(300,30)(100, 10), (200, 20), (300, 30) and join them to get a straight line passing through the origin.
  4. To find interest for Rs.250Rs. 250, locate 250250 on the xx-axis, move vertically to meet the graph, then horizontally to the yy-axis.
  5. The yy-value corresponds to 2525. So, interest = Rs.25Rs. 25.

Explanation:

This is a linear graph problem where we use the coordinates to map the relationship between principal and interest. The graph allows us to interpolate values like Rs.250Rs. 250 that weren't in our initial data set.

Problem 2:

Identify the coordinates of a point AA which lies 55 units to the left of the yy-axis and 22 units below the xx-axis. Also, identify which quadrant it lies in.

Solution:

  1. '5 units to the left of the yy-axis' means the xx-coordinate is −5-5.
  2. '2 units below the xx-axis' means the yy-coordinate is −2-2.
  3. Combining these, the coordinates of point AA are (−5,−2)(-5, -2).
  4. Since both xx and yy are negative, the point lies in the Third Quadrant.

Explanation:

To find coordinates, we translate directional descriptions into positive or negative values relative to the origin. Left and Down signify negative values in the Cartesian system.