krit.club logo

Introduction to Graphs - Reading and Interpreting Graphs

Grade 8CBSE

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

🔑Concepts

•

Graphs provide a visual representation of data, making it easier to see patterns and trends. The most fundamental system is the Cartesian coordinate system, which uses two perpendicular lines called axes: the horizontal xx-axis and the vertical yy-axis.

Cartesian coordinate system showing x and y axes with a point P(3,2).
•

A line graph consists of segments joined together to show trends over time. When the graph is a single continuous straight line, it is called a linear graph.

A straight line graph passing through the origin representing a linear relationship.
•

Reading a graph involves finding the value of one variable for a given value of another. For example, to find the yy-value at x=4x = 4, move vertically from x=4x = 4 to the graph, then horizontally to the yy-axis.

•

Independent variables (the cause) are usually plotted on the horizontal xx-axis, while dependent variables (the effect) are plotted on the vertical yy-axis.

📐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.

Problem 3:

A car travels at a uniform speed of 40 km/h40\text{ km/h}. Draw a distance-time graph for this motion and find the distance covered by the car in 2.52.5 hours.

Distance-time graph showing a straight line where x=2.5 corresponds to y=100.

Solution:

  1. Create a table of values: Time tt (h) vs Distance dd (km). At t=0,d=0t=0, d=0; At t=1,d=40t=1, d=40; At t=2,d=80t=2, d=80.
  2. Plot the points (0,0),(1,40),(2,80)(0,0), (1,40), (2,80) and join them with a straight line.
  3. To find distance at t=2.5t=2.5 hours, locate 2.52.5 on the xx-axis, move up to the line, and read the yy-value. Distance = 40×2.5=100 km40 \times 2.5 = 100\text{ km}.

Explanation:

Since the speed is uniform, the distance is directly proportional to time (d=40td = 40t), resulting in a linear graph.

Problem 4:

The following graph shows the temperature of a patient in a hospital, recorded every hour. At what time was the patient's temperature 38.5∘C38.5^{\circ}\text{C}?

A line graph where a horizontal line from 38.5 on the y-axis meets the graph at x=11.

Solution:

  1. Look at the yy-axis (Temperature) and find the mark for 38.5∘C38.5^{\circ}\text{C}.
  2. Move horizontally from 38.538.5 until you intersect the graph.
  3. From that point of intersection, move vertically down to the xx-axis (Time).
  4. The corresponding time on the xx-axis is 11:00 am11:00\text{ am}.

Explanation:

To interpret a specific value, we use the intersection of horizontal and vertical projections from the axes to the plotted line.