Drawing and Reading Graphs ഗ്രാഫ്
Half of this chapter is about graphs (ഗ്രാഫ്): the road-accident graph, position–time graphs, velocity–time graphs, and the many graph questions in “Let us assess”. A graph is a picture of a table of numbers. Once you can draw one and read one, you can see the whole story of a motion at a glance. This page teaches you to do both, step by step, on graph paper.
- Positive and negative numbers on a number line (page 1).
- Multiplying and dividing simple numbers.
1. Axes, origin and the four quarters
A graph is drawn on graph paper, which is ruled into 1 cm squares, and each of those into small 1 mm squares. Two number lines are drawn on it at right angles:
- the horizontal line XX′ is the X-axis (X അക്ഷം);
- the vertical line YY′ is the Y-axis (Y അക്ഷം);
- the point O where they cross is the origin (മൂലബിന്ദു), the zero of both lines.
Along OX (to the right) the numbers are positive, and along OX′ (to the left) they are negative. Along OY (up) they are positive, and along OY′ (down) negative. The two axes divide the paper into four quarters, called quadrants I, II, III and IV. Any point is named by an ordered pair (x, y): first how far across, then how far up or down. A 1 cm × 1 cm square such as ABCD has an area (പരപ്പളവ്) of 1 cm².
Always read the x value first (along the X-axis), then the y value. (3, 2) and (2, 3) are different points! The blue square ABCD is 1 cm × 1 cm = 1 cm², like the square in the textbook's Graph 2.2.
A graph is named after its two quantities, the Y quantity first: “Y quantity – X quantity graph”. Position on the Y-axis and time on the X-axis gives a position–time graph (സ്ഥാന-സമയ ഗ്രാഫ്). Velocity on Y and time on X gives a velocity–time graph (പ്രവേഗ-സമയ ഗ്രാഫ്). Time almost always goes on the X-axis. In motion graphs we only need quadrant I, because time and (usually) position are positive.
2. Plotting a table, point by point
Suppose a girl cycles past the electric posts on a village road, and her friend writes down her position every 2 seconds:
| Time (s) – X-axis | 0 | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|---|
| Position (m) – Y-axis | 0 | 5 | 10 | 15 | 20 | 25 |
Each column is one ordered pair: (0, 0), (2, 5), (4, 10) and so on. Watch how to put them on graph paper:
3. Choosing a scale
The scale (തോത്) says how much one centimetre of graph paper stands for, for example X-axis: 1 cm = 2 s. You choose it yourself, for each axis separately. A good scale:
- makes the largest value fit on the paper;
- makes the graph fill most of the paper, so it is easy to read;
- uses easy steps such as 1, 2, 5 or 10 units per cm, so that the in-between values are easy to find.
Always write the scale in a corner of the graph, as the textbook does.
The data: a car's velocity is 0, 8, 16, 24, 32 and 40 m/s at 0, 2, 4, 6, 8 and 10 s. The paper is 12 cm wide and 9 cm tall. Try every scale. Which one fits and fills the paper best?
Change the scale and the line looks steeper or flatter. But the values it shows, and anything you calculate from them, stay exactly the same. In the lab, the car always gains 40 m/s in 10 s, so its acceleration is always 4 m/s².
4. Reading values from a graph
To read a graph, go up from a value on the X-axis to the line, then across to the Y-axis, and read the number there (using the scale). To go the other way, start on the Y-axis, go across to the line, then down to the X-axis.
Red guide: from a time, read the velocity. Blue guide: from a velocity, find the time. Scale: X 1 cm = 1 s, Y 1 cm = 5 m/s.
Worked example 1 · Using the scale
Q. A velocity–time graph is drawn with the scale X-axis: 1 cm = 2 s and Y-axis: 1 cm = 10 m/s. A point on it is 3 cm to the right of O and 2.5 cm above O. What time and what velocity does this point show?
Value = (number of cm) × (what 1 cm stands for). Do it once for each axis.
5. Straight lines and curves
When you join the points, the line can be a straight line (നേർരേഖ), a sloping straight line (ചരിവുള്ള നേർരേഖ) or a curve (വക്രരേഖ). Use a ruler only when the points really lie on a straight line; otherwise draw a smooth curve through them freehand. The shape tells you how the Y quantity changes as the X quantity grows:
The small steps under the line show how much Y grows for each 1 cm across. Equal steps give a straight line; changing steps give a curve.
6. Slope: rise ÷ run
The slope of a line tells how steep it is. Pick two points A and B on the line. The rise is how much the Y value goes up between them, and the run is how much the X value goes across:
Always use the values with units read from the axes, not the centimetres. Then the slope has a unit too. On a position–time graph the rise is metres and the run is seconds, so the slope is in m/s: it is the velocity! On a velocity–time graph the slope is in m/s per s = m/s²: the acceleration. You will use both on page 10.
Move A and B anywhere on the line: the triangle grows or shrinks, but rise ÷ run stays the same. A straight line has the same slope everywhere. That is why a straight position–time line means a uniform velocity.
Worked example 2 · Finding a slope
Q. A position–time graph is a straight line passing through A (2 s, 6 m) and B (8 s, 24 m). Find the slope of the line. What does it tell you?
Rise = 24 − 6 (metres). Run = 8 − 2 (seconds). Slope = rise ÷ run, with the unit m/s.
Kerala Police publish the number of road accidents every year, and the textbook shows them as a graph (Graph 2.1). One glance at the line shows which year had the fewest accidents and how fast the number rose afterwards, which is much quicker than reading a table. In the school lab, the ExpEYES kit can measure the distance of a moving object with an echo module and plot its position–time graph on a computer while the object moves.