9 min readPathway sheetPart IV graph (8 marks), often a word problem; the shape of the graph comes as a 1-mark MCQ

Graphs of direct and inverse variation

Direct variation y = kx gives a straight line through the origin. Inverse variation xy = k gives a smooth curve that never touches the axes.

Direct variationy = kx, so the ratio y/x stays the same

Inverse variationxy = k, so the product stays the same

Direct graphStraight line through the origin (0, 0)

Inverse graphSmooth curve (one branch of a rectangular hyperbola) that never meets the axes

Direct vs inverse variation

Direct variationWhen one quantity increases, the other increases in the same ratio. y = kx, where k = y/x is the constant of variation. Example: shirts stitched and hours worked.

Inverse variationWhen one quantity increases, the other decreases in the same ratio. xy = k. Example: speed and time for a fixed journey.

Test a table: if y/x is the same for every pair, it is direct variation. If xy is the same, it is inverse variation.

How to draw the graph: direct variation

Problem: a machine at a Tiruppur garment unit stitches 6 shirts every hour. Draw the graph, and find (i) the shirts stitched in 4½ hours (ii) the time taken to stitch 21 shirts.

1
Relation: shirts y = 6 × hours x, so y = 6x and k = 6. This is direct variation.
2
Table of values: x = 1, 2, 3, 4, 5, 6 gives y = 6, 12, 18, 24, 30, 36.
3
Scale: x-axis 2 cm = 1 hour; y-axis 1 cm = 3 shirts. Write the scale in the corner of the sheet.
4
Plot the points (1, 6), (2, 12), (3, 18), (4, 24), (5, 30) and (6, 36).
5
Join them with a ruler. The line passes through the origin O, since 0 hours gives 0 shirts.
6★
Read (i): from x = 4.5 go up to the line, then across to the y-axis: y = 27 shirts.
7★
Read (ii): from y = 21 go across to the line, then down to the x-axis: x = 3.5 hours.
8
Check: 6 × 4.5 = 27 and 21 ÷ 6 = 3.5.

★ key step

How to draw the graph: inverse variation

Problem: a 60 km road joins a town to a hill temple. Vehicles take different times at different speeds. Draw the speed-time graph, and find (i) the time taken at 40 km/h (ii) the speed needed to take 5 hours.

1
Relation: speed x × time y = 60, so xy = 60 and k = 60. This is inverse variation.
2
Table of values: x = 10, 15, 20, 30, 60 km/h gives y = 6, 4, 3, 2, 1 hours.
3
Scale: x-axis 1 cm = 5 km/h; y-axis 2 cm = 1 hour.
4
Plot the points (10, 6), (15, 4), (20, 3), (30, 2) and (60, 1).
5
Join them with a smooth free-hand curve. It bends towards the axes but never touches them.
6★
Read (i): from x = 40 go up to the curve, then across to the y-axis: y = 1.5 hours.
7★
Read (ii): from y = 5 go across to the curve, then down to the x-axis: x = 12 km/h.
8
Check: 40 × 1.5 = 60 and 12 × 5 = 60.

★ key step

Worked example: direct or inverse?

Problem

Table A: a tap runs for x hours = 1, 2, 3, 4 and the tank holds y litres = 40, 80, 120, 160. Table B: x equal taps = 1, 2, 3, 4, 6 fill a tank in y minutes = 60, 30, 20, 15, 10. Name the variation in each table, find k, and find y when x = 5.

Working

Table A: y/x = 40/1 = 80/2 = 120/3 = 160/4 = 40. The ratio is constant, so A is direct variation with k = 40.
A: y = 40x. When x = 5, y = 40 × 5 = 200 litres.
Table B: xy = 1 × 60 = 2 × 30 = 3 × 20 = 4 × 15 = 6 × 10 = 60. The product is constant, so B is inverse variation with k = 60.
B: xy = 60. When x = 5, y = 60 ÷ 5 = 12 minutes.

Answer

A: direct, k = 40, y = 200 litres. B: inverse, k = 60, y = 12 minutes.

Graph sheet habits that earn marks

Write the scale in the top right corner of the sheet.
Label each axis with its quantity and unit, such as 'Time (hours)'.
Use a ruler for a direct variation line and a smooth free-hand curve for inverse variation.
Show each reading with dotted lines from the axis to the graph and on to the other axis.
For real quantities like speed and time, plot only positive values, in the first quadrant.
End with a sentence that begins 'From the graph, ...'.

In real life

A shop charging the same rate per kg is direct variation: double the rice, double the bill. Sharing a fixed job among more workers is inverse variation: more workers, fewer days.

Exam angle

1 marks: “What is the shape of the graph of xy = k (k > 0) for x > 0 and y > 0?” A smooth curve, one branch of a **rectangular hyperbola**, that comes closer and closer to the axes but never meets them.

8 marks: “Draw the graph of xy = 12 for x > 0, y > 0. From the graph find (i) y when x = 5 (ii) x when y = 8.” Table: x = 1, 2, 3, 4, 6, 12 gives y = 12, 6, 4, 3, 2, 1. Scale 1 cm = 1 unit on both axes. Plot and join with a smooth curve. Read: (i) y = 2.4 when x = 5 (ii) x = 1.5 when y = 8. Check: 5 × 2.4 = 12 and 1.5 × 8 = 12.

8 marks: “A school canteen sells idlis at ₹ 5 each (an example price). Draw the graph of cost against the number of idlis, from 1 to 8. From the graph find the cost of 7 idlis and the number of idlis bought for ₹ 30.” Relation y = 5x, so k = 5. Table: x = 1 to 8 gives y = 5 to 40. Scale: x-axis 1 cm = 1 idli, y-axis 1 cm = ₹ 5. Straight line through O. Read: 7 idlis cost ₹ 35; ₹ 30 buys 6 idlis.

Remember it

Direct: same ratio, straight line, through O. Inverse: same product, curved, never touches the axes.