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Introduction to Graphs - Line Graphs

Grade 8ICSE

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

🔑Concepts

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A line graph is a type of chart used to visualize data that changes continuously over a period of time. It consists of several data points called 'markers' which are connected by straight line segments to show a trend or pattern. Visually, this appears as a series of connected dots forming a jagged or straight path across a grid.

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The graph is constructed on a Cartesian Plane consisting of two perpendicular lines: the horizontal line is called the xx-axis (abscissa) and the vertical line is called the yy-axis (ordinate). The point where these two axes intersect is called the Origin, denoted by O(0,0)O(0, 0).

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To represent data, we use ordered pairs (x,y)(x, y). The first number, the xx-coordinate, indicates the horizontal distance from the yy-axis, and the second number, the yy-coordinate, indicates the vertical distance from the xx-axis. For example, the point (3,5)(3, 5) is 3 units to the right of the origin and 5 units up.

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Selecting a proper scale is crucial for clarity. The scale is the value assigned to each unit or division on the graph paper (e.g., 1 cm=5 units1\text{ cm} = 5\text{ units}). If the data values start far from zero, a 'kink' or 'break' (represented by a zigzag line on the axis) is used to indicate that the scale does not start from the origin.

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Line graphs distinguish between Independent and Dependent variables. The independent variable (like time or years) is usually plotted on the xx-axis, while the dependent variable (like temperature, profit, or distance), which changes based on the independent variable, is plotted on the yy-axis.

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Interpreting the slope of the line segments provides information about the data: an upward-sloping line indicates an increase, a downward-sloping line indicates a decrease, and a horizontal line indicates that the value remained constant over that specific interval.

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A Linear Graph is a specific type of line graph where all the plotted points lie on a single straight line. This signifies a direct and proportional relationship between the two variables, represented by the equation y=mx+cy = mx + c.

📐Formulae

Coordinates of a point: (x,y)(x, y) where xx is the abscissa and yy is the ordinate.

General equation of a straight line (Linear Graph): y=mx+cy = mx + c, where mm is the slope and cc is the yy-intercept.

Slope (mm) between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2): m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Midpoint Formula: M=(x1+x22,y1+y22)M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})

Distance between two points P(x1,y1)P(x_1, y_1) and Q(x2,y2)Q(x_2, y_2): d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

💡Examples

Problem 1:

A car travels at a constant speed of 40 km/h40\text{ km/h}. Create a distance-time table for 0,1,2,30, 1, 2, 3 hours and describe how to plot the resulting linear graph.

Solution:

Step 1: Create the table based on the formula Distance=Speed×Time\text{Distance} = \text{Speed} \times \text{Time}.

  • At t=0 h,d=40×0=0 kmt = 0\text{ h}, d = 40 \times 0 = 0\text{ km}. Point: (0,0)(0, 0)
  • At t=1 h,d=40×1=40 kmt = 1\text{ h}, d = 40 \times 1 = 40\text{ km}. Point: (1,40)(1, 40)
  • At t=2 h,d=40×2=80 kmt = 2\text{ h}, d = 40 \times 2 = 80\text{ km}. Point: (2,80)(2, 80)
  • At t=3 h,d=40×3=120 kmt = 3\text{ h}, d = 40 \times 3 = 120\text{ km}. Point: (3,120)(3, 120)

Step 2: On a graph paper, take Time on the xx-axis (1 unit=1 hour1\text{ unit} = 1\text{ hour}) and Distance on the yy-axis (1 unit=20 km1\text{ unit} = 20\text{ km}). Step 3: Plot the points (0,0),(1,40),(2,80),(3,120)(0,0), (1,40), (2,80), (3,120) and join them with a straight line.

Explanation:

Since the speed is constant, the relationship between distance and time is linear. The graph will be a straight line passing through the origin, showing that distance is directly proportional to time.

Problem 2:

The following coordinates represent the temperature recorded at different times of the day: (6 AM,15∘C),(9 AM,20∘C),(12 PM,25∘C),(3 PM,22∘C)(6\text{ AM}, 15^{\circ}\text{C}), (9\text{ AM}, 20^{\circ}\text{C}), (12\text{ PM}, 25^{\circ}\text{C}), (3\text{ PM}, 22^{\circ}\text{C}). Find the increase in temperature between 6 AM6\text{ AM} and 12 PM12\text{ PM}.

Solution:

Step 1: Identify the yy-coordinates (temperatures) for the given times.

  • Temperature at 6 AM(y1)=15∘C6\text{ AM} (y_1) = 15^{\circ}\text{C}
  • Temperature at 12 PM(y2)=25∘C12\text{ PM} (y_2) = 25^{\circ}\text{C}

Step 2: Calculate the difference (increase). Increase=y2−y1\text{Increase} = y_2 - y_1 Increase=25∘C−15∘C=10∘C\text{Increase} = 25^{\circ}\text{C} - 15^{\circ}\text{C} = 10^{\circ}\text{C}

Explanation:

By reading the yy-values corresponding to the specific points on the xx-axis (time), we can determine the change in the dependent variable (temperature) over the specified interval.