Class 8 Maths Notes Introduction to Graphs

1. What is a graph?

A visual representation of numerical data.

2. Name a graph used to compare parts of a whole.

Pie graph (or Circle graph).

3. Name a graph used to compare categories.

Bar graph.

4. Which graph uses bars of uniform width with equal spacing?

Bar graph.

5. Which graph uses bars adjacent to each other (no gaps)?

Histogram.

6. What kind of data is usually represented by a histogram?

Grouped data with continuous class intervals.

7. Which graph is used to show changes over time?

Line graph.

8. What is a linear graph?

A line graph which forms a single straight line.

9. What is the horizontal axis called in a coordinate system?

x-axis.

10. What is the vertical axis called?

y-axis.

11. What are the coordinates of the Origin?

(0, 0).

12. In the coordinate (3, 5), what is the x-coordinate (abscissa)?

3.

13. In the coordinate (3, 5), what is the y-coordinate (ordinate)?

5.

14. On which axis does the point (0, 4) lie?

y-axis (since x=0).

15. On which axis does the point (5, 0) lie?

x-axis (since y=0).

16. In a pie chart, what is the sum of all central angles?

360°.

17. If a sector represents 50% in a pie chart, what is its central angle?

50% of 360° = (50/100) * 360° = 180°.

1. Differentiate between a bar graph and a histogram.

• Bar Graph: Compares discrete categories. Bars have uniform width and equal spacing between them.
• Histogram: Shows frequency distribution of continuous data in class intervals. Bars represent intervals and are drawn adjacent to each other (no gaps).

2. What information does a pie chart give? How do you calculate the central angle for a component?

• A pie chart shows the relationship between a whole and its parts (proportion/percentage of each category).
• The whole circle represents 100% or 360°.
• Central angle for a component = (Value of the component / Total value) × 360°.
• Or, Central angle = (Percentage of component / 100) × 360°.

3. When is it appropriate to use a line graph?

• Line graphs are best used to show data that changes continuously over a period of time.
• They effectively display trends, like increase or decrease.
• Examples: Temperature changes during a day, patient’s pulse rate over time, yearly sales figures.

4. What are coordinates? Explain how to locate the point (2, -3).

• Coordinates are a pair of numbers (x, y) that specify the exact location of a point on a coordinate plane.
• ‘x’ is the distance along the x-axis (horizontal).
• ‘y’ is the distance along the y-axis (vertical).
• To locate (2, -3):
  • Start at Origin (0,0).
  • Move 2 units Right along x-axis (because x=+2).
  • From there, move 3 units Down parallel to y-axis (because y=-3). Mark the point.

5. Does the point (4, 1) lie on the line passing through (2, 3) and (6, -1)? (Hint: Find slope or equation – maybe too advanced? Alternative: Check collinearity conceptually or using distance – also hard. Class 8 way: Just state plotting determines this).

• To check if points are on the same line (collinear) without using slope/equation formula (advanced for Class 8 usually):
• You would plot all three points accurately on graph paper.
• Visually check if the three points lie on a single straight line that can be drawn with a ruler.
• *Explanation:* If points (x₁, y₁), (x₂, y₂), (x₃, y₃) are collinear, the slope between (x₁, y₁) and (x₂, y₂) must be the same as the slope between (x₂, y₂) and (x₃, y₃). Slope = (change in y) / (change in x).
• Slope(A,B) = (-1-3)/(6-2) = -4/4 = -1. Slope(B,C) = (1 – (-1))/(4 – 6) = 2 / (-2) = -1. *Self-correction: Since the slopes are the same, yes, the point (4, 1) lies on the line passing through (2, 3) and (6, -1).* [Adapted answer to show the concept without assuming advanced student knowledge directly.]

6. What is the x-coordinate also known as? What is the y-coordinate also known as?

• x-coordinate is also called the Abscissa.
• y-coordinate is also called the Ordinate.

7. Draw a rough sketch of a possible line graph showing a car’s speed decreasing uniformly over time.

• Draw x-axis (Time) and y-axis (Speed).
• Plot points representing speed at different times.
• Since speed decreases uniformly, the points should form a straight line sloping downwards from left to right.
• Connect the points with a straight line.

(Imagine a graph with Time increasing on the bottom, Speed increasing upwards, and a line starting high on the left and ending low on the right).

8. For which situation would a Pie Chart be more suitable than a Bar Graph?

• A Pie Chart is more suitable when you want to show the **proportion** or **percentage** contribution of different parts to a whole total.
• Example: Percentage distribution of favourite sports among students in a class (Total students = 100%).
• A Bar Graph is better for comparing the **actual values** or frequencies of different independent categories.
• Example: Comparing the number of students who like Cricket vs Football vs Basketball.

9. Plot the points A(4, 0), B(4, 2), C(4, 6), D(4, 2.5) on a graph paper. Do they lie on a line? If yes, describe it.

• Draw x and y axes. Mark points 0, 1, 2… on both.
• Plot A(4,0): Move 4 right on x-axis, 0 up. (On x-axis).
• Plot B(4,2): Move 4 right, 2 up.
• Plot C(4,6): Move 4 right, 6 up.
• Plot D(4,2.5): Move 4 right, 2.5 up (between 2 and 3 on y scale).
• Yes, all points lie on a single **vertical line**.
• This line is parallel to the y-axis and passes through x = 4.

10. The following graph shows the temperature of a patient. (a) What was the temp at 1 pm? (b) When was temp highest? [Conceptual Q]

(Assuming a typical line graph for patient temperature)

• (a) To find temp at 1 pm: Locate ‘1 pm’ on the x-axis (Time). Move vertically up to the graph line. Then move horizontally left to read the corresponding value on the y-axis (Temperature).
• (b) To find when temp was highest: Look for the highest point reached by the graph line. Read the corresponding time on the x-axis below this highest point.

11. Can there be a time-temperature graph as follows? Justify. [Conceptual: Graph shows temp increasing then decreasing, then suddenly vertical jump up, then continues].

• The graph showing temperature changing over time (increasing, decreasing) is possible.
• However, a sudden **vertical jump** in temperature at a single point in time is **not possible**.
• A vertical line on a time-temperature graph would mean the patient had multiple different temperatures at the exact same instant, which is physically impossible.
• Therefore, such a graph cannot represent a real situation accurately.

1. Explain the different types of graphs studied: Bar graph, Pie graph, Histogram, and Line graph. Mention what each is typically used for.

• 1. Bar Graph: Uses rectangular bars of uniform width with equal spacing to compare discrete items/categories. Height/length represents value. Use: Comparing marks in different subjects, production in different years.
• 2. Pie Graph (Circle Graph): Represents parts of a whole as sectors (slices) of a circle. Size of sector proportional to value/percentage. Use: Showing percentage distribution of budget, survey results, time spent on activities.
• 3. Histogram: Like bar graph but for continuous data grouped into class intervals. Bars touch each other (no gaps). Height represents frequency in each interval. Use: Showing distribution of heights/weights in different ranges, marks in different intervals.
• 4. Line Graph: Shows data changing continuously over time using points connected by lines. Use: Tracking temperature change, stock prices over time, patient’s pulse rate.

2. Describe the Cartesian coordinate system used for plotting graphs.

The Cartesian system helps locate points on a plane:

• Axes: Uses two perpendicular number lines intersecting at a point.
  • x-axis: The horizontal number line.
  • y-axis: The vertical number line.
• Origin (O): The point where the x-axis and y-axis intersect. Its coordinates are (0, 0).
• Coordinates (x, y): An ordered pair describing a point’s location.
  • x-coordinate (Abscissa): The perpendicular distance from the y-axis (measured along x-axis; right is positive, left is negative).
  • y-coordinate (Ordinate): The perpendicular distance from the x-axis (measured along y-axis; up is positive, down is negative).
• Quadrants: The axes divide the plane into four regions called quadrants (I, II, III, IV) based on the signs of x and y coordinates.
• Plotting: To plot (x, y), start at origin, move x units horizontally, then y units vertically.

3. What is a linear graph? Plot the points (1, 2), (2, 4), (3, 6), (4, 8). Do they lie on a line? Can you write the linear equation for this?

Linear Graph: A graph formed by plotting points that represent a linear relationship between two variables, such that all the points lie on a single straight line.

Plotting Points:

• (1, 2): 1 right, 2 up from origin.
• (2, 4): 2 right, 4 up from origin.
• (3, 6): 3 right, 6 up from origin.
• (4, 8): 4 right, 8 up from origin.

Observation:

• Yes, when plotted on graph paper, these points will all lie on a single straight line passing through the origin (0,0 – although not plotted here).

Linear Equation:

• Observe the relationship: The y-coordinate is always twice the x-coordinate (y = 2 × x).
• The linear equation representing this relationship is y = 2x.

4. Draw a line graph for the following data showing the temperature forecast and the actual temperature for each day of a week.

(Data Table Conceptual Example)

Steps to Draw Line Graph:

1. Draw horizontal x-axis (label ‘Day’) and vertical y-axis (label ‘Temperature (°C)’).
2. Mark days (Mon, Tue…) equally spaced on x-axis.
3. Choose a suitable scale for y-axis (e.g., starting from 24, mark 25, 26, 27, 28).
4. Plot points for Forecast temperature for each day. Join these points with a dotted or dashed line (or one color). Label this line ‘Forecast’.
5. Plot points for Actual temperature for each day. Join these points with a solid line (or another color). Label this line ‘Actual’.
6. Add a Title to the graph (e.g., “Comparison of Forecast vs Actual Temperature”).
7. Include a Key/Legend explaining which line represents Forecast and which represents Actual.

(Description of graph appearance: Two lines would be seen, possibly close together sometimes, diverging at other times like on Thursday/Friday, showing the comparison visually).

5. A courier person cycles from town A to town B. His distance from town A at different times is shown. Plot a time-distance graph.

(Data Table Conceptual Example)

Steps to Draw Time-Distance Graph:

1. Draw x-axis (Time, starting from 8:00 AM) and y-axis (Distance in km, starting from 0).
2. Mark times (8:00, 9:00, 10:00, 11:00 AM) equally spaced on x-axis.
3. Choose a suitable scale for y-axis (e.g., 1 cm = 5 km, marking 0, 5, 10, 15, 20, 25, 30).
4. Plot the points corresponding to the data: (8:00, 0), (9:00, 10), (10:00, 20), (11:00, 30).
5. Join the plotted points using line segments.

Observation: The points lie on a straight line. This indicates the courier person is cycling at a constant speed.

6. When is a Pie chart more useful than a Histogram?

• A Pie Chart is more useful when the primary goal is to show the **proportion or percentage** of different categories that make up a whole. It emphasizes the relationship of each part to the total. 📊➡️💯 (Jab humein poore ka kitna hissa ya percentage har category hai, yeh dikhana ho).
• A Histogram is more useful when you want to see the **frequency distribution** of continuous data grouped into intervals, showing how often values fall within specific ranges. It shows the shape of the data distribution. (Jab humein dekhna ho ki data alag-alag range mein kitni baar aata hai).
• Example where Pie Chart is better: Showing breakdown of expenses (Food%, Rent%, Travel% etc.) from a total monthly income.
• Example where Histogram is better: Showing the number of students scoring marks in ranges 0-10, 10-20, 20-30 etc. in an exam.

7. Describe how you would locate the coordinates of a given point P on a graph paper.

To find the coordinates (x, y) of a point P:

1. Find x-coordinate (Abscissa): Draw a perpendicular line from point P down to the x-axis (or up, if P is below). The value where this perpendicular meets the x-axis is the x-coordinate. Alternatively, count the horizontal distance from the y-axis to point P (right is positive, left is negative). (P se x-axis par perpendicular daalo. Jahaan woh x-axis ko mile, woh x-coordinate hai).
2. Find y-coordinate (Ordinate): Draw a perpendicular line from point P across to the y-axis (left or right). The value where this perpendicular meets the y-axis is the y-coordinate. Alternatively, count the vertical distance from the x-axis to point P (up is positive, down is negative). (P se y-axis par perpendicular daalo. Jahaan woh y-axis ko mile, woh y-coordinate hai).
3. Write the Coordinates: Write the coordinates as an ordered pair (x, y), with the x-coordinate first, followed by the y-coordinate, enclosed in parentheses and separated by a comma.

8. Plot the following points on a graph sheet and check if they lie on a straight line: K(2, 3), L(5, 3), M(5, 5), N(2, 5).

Steps:

1. Draw x and y axes on graph paper. Mark points (0, 1, 2, …).
2. Plot K(2, 3): Move 2 right, 3 up. Mark K.
3. Plot L(5, 3): Move 5 right, 3 up. Mark L.
4. Plot M(5, 5): Move 5 right, 5 up. Mark M.
5. Plot N(2, 5): Move 2 right, 5 up. Mark N.

Observation:

• Join the points K, L, M, N in order using a ruler.
• You will see that KL, LM, MN, and NK form straight line segments.
• The points K, L, M, N themselves do **NOT** all lie on a single straight line.
• They form the vertices of a shape. KL is horizontal (y=3), LM is vertical (x=5), MN is horizontal (y=5), NK is vertical (x=2).
• The shape formed is a **Rectangle** (or a square if scales were equal and looked square, but coordinates confirm rectangle).

9. What kind of information does a histogram provide compared to a bar graph?

While both use bars, they show different information:

• Bar Graph: Shows comparison between **discrete categories**. Height of bar shows value/frequency for *that specific category*. There are gaps between bars, indicating categories are distinct. (Alag-alag categories (jaise subjects, years) ki tulna karta hai. Bars ke beech gap hota hai).
• Histogram: Shows the **frequency distribution** of **continuous data** grouped into **class intervals**. Height of bar shows the frequency of data points falling *within that specific range/interval*. There are no gaps between bars, indicating the data is continuous across intervals. (Lagatar data (jaise height range) ka distribution dikhata hai. Bars ke beech gap nahi hota).
• Essentially, bar graphs compare individual items, while histograms show how data is spread out over a continuous range.

10. Give three real-life examples where line graphs are commonly used.

Line graphs are used to show trends or changes over time:

1. Weather Reports 🌡️: Showing the change in temperature throughout a day or across several days/weeks. Allows seeing trends like warming up or cooling down.
2. Stock Market 📈📉: Tracking the price of a stock or index (like Sensex/Nifty) over days, months, or years to see its performance and trends.
3. Medical Monitoring ❤️‍🩹: Plotting a patient’s temperature, heart rate, or blood pressure recorded at regular intervals to monitor their health status and see changes.
4. (Other examples: Population growth over decades, Cricket match run rate over overs, Sales figures of a company over quarters).