Graphs of Direct and Inverse Proportions
When two quantities are in direct proportion, their graph is a straight line passing through the origin. When two quantities are in inverse proportion, their graph is a curve (hyperbola) that never touches the axes.
Graphs give a visual way to identify the type of proportion between quantities. By plotting values on a coordinate plane, you can quickly see whether the relationship is direct, inverse, or neither.
In this topic, you will learn how to plot graphs for direct and inverse proportions and how to read information from these graphs.
What is Graphs of Direct and Inverse Proportions?
Definition: The graph of direct proportion is a straight line through the origin (0, 0), with equation y = kx.
Definition: The graph of inverse proportion is a rectangular hyperbola with equation y = k/x or xy = k.
Direct: y = kx (straight line)
Inverse: xy = k (hyperbola)
Where:
- k = constant of proportionality
- In direct proportion, y/x = k (constant)
- In inverse proportion, xy = k (constant)
Methods
How to plot a graph of direct proportion:
- Make a table of x and y values where y = kx.
- Plot the points on graph paper.
- Join the points — they form a straight line through the origin.
How to plot a graph of inverse proportion:
- Make a table of x and y values where xy = k.
- Plot the points on graph paper.
- Join the points with a smooth curve — this is a hyperbola.
- The curve approaches but never touches the x-axis or y-axis.
How to identify the type from a graph:
- Straight line through origin → Direct proportion
- Straight line NOT through origin → Linear but not proportional
- Curve approaching axes → Inverse proportion
Solved Examples
Example 1: Example 1: Plotting direct proportion
Problem: Plot the graph for y = 3x using values x = 1, 2, 3, 4, 5.
Solution:
Table of values:
- x = 1, y = 3
- x = 2, y = 6
- x = 3, y = 9
- x = 4, y = 12
- x = 5, y = 15
Plot these points: (1,3), (2,6), (3,9), (4,12), (5,15).
Join them — you get a straight line through the origin.
Answer: The graph is a straight line through (0,0) with slope 3.
Example 2: Example 2: Plotting inverse proportion
Problem: Plot the graph for xy = 12 using values x = 1, 2, 3, 4, 6, 12.
Solution:
Table of values (y = 12/x):
- x = 1, y = 12
- x = 2, y = 6
- x = 3, y = 4
- x = 4, y = 3
- x = 6, y = 2
- x = 12, y = 1
Plot these points and join with a smooth curve.
Answer: The graph is a curve (hyperbola) that approaches but never touches the axes.
Example 3: Example 3: Identifying from a graph
Problem: A graph passes through the points (2, 6), (4, 12), and (5, 15). What type of proportion is this?
Solution:
- Check y/x ratio: 6/2 = 3, 12/4 = 3, 15/5 = 3
- The ratio is constant (k = 3).
- Also check: does it pass through (0, 0)? When x = 0, y = 3(0) = 0. Yes.
Answer: This is direct proportion with k = 3.
Example 4: Example 4: Distance-time (direct)
Problem: A car travels at 60 km/h. Plot the distance-time graph for t = 1, 2, 3, 4 hours.
Solution:
d = 60t (direct proportion):
- t = 1 h, d = 60 km
- t = 2 h, d = 120 km
- t = 3 h, d = 180 km
- t = 4 h, d = 240 km
The graph is a straight line through the origin with slope 60.
Answer: Distance and time are in direct proportion.
Example 5: Example 5: Speed-time (inverse)
Problem: A journey is 120 km. Plot the speed-time graph for speeds 20, 30, 40, 60 km/h.
Solution:
time = 120/speed (inverse proportion):
- Speed = 20, Time = 6 h
- Speed = 30, Time = 4 h
- Speed = 40, Time = 3 h
- Speed = 60, Time = 2 h
The graph is a curve — as speed increases, time decreases.
Answer: Speed and time are in inverse proportion (for fixed distance).
Example 6: Example 6: Reading values from a direct proportion graph
Problem: The graph of y = 2.5x is given. Find y when x = 6.
Solution:
- y = 2.5 × 6 = 15
- On the graph, locate x = 6 on the horizontal axis, go up to the line, and read y = 15.
Answer: When x = 6, y = 15.
Example 7: Example 7: Workers and days (inverse)
Problem: 12 workers finish a job in certain days. The product (workers × days) = 60. Plot the graph.
Solution:
- Workers = 2, Days = 30
- Workers = 3, Days = 20
- Workers = 4, Days = 15
- Workers = 5, Days = 12
- Workers = 6, Days = 10
- Workers = 10, Days = 6
The graph is a hyperbola (curve).
Answer: Workers and days are in inverse proportion.
Example 8: Example 8: Verifying from a table
Problem: x: 2, 4, 5, 10 and y: 20, 10, 8, 4. Is this direct or inverse?
Solution:
- Check y/x: 20/2 = 10, 10/4 = 2.5 — NOT constant. Not direct.
- Check xy: 2×20 = 40, 4×10 = 40, 5×8 = 40, 10×4 = 40 — constant!
Answer: This is inverse proportion with k = 40. The graph would be a hyperbola.
Example 9: Example 9: Cost and quantity (direct)
Problem: Apples cost Rs 80 per kg. Plot a graph for 1, 2, 3, 4, 5 kg.
Solution:
- 1 kg = Rs 80, 2 kg = Rs 160, 3 kg = Rs 240, 4 kg = Rs 320, 5 kg = Rs 400
- Ratio: cost/kg = 80 (constant)
The graph is a straight line through the origin.
Answer: Cost and quantity are in direct proportion.
Example 10: Example 10: Identifying non-proportional data
Problem: x: 1, 2, 3, 4 and y: 3, 5, 7, 9. Is this direct proportion?
Solution:
- y/x: 3/1 = 3, 5/2 = 2.5, 7/3 = 2.33... — NOT constant.
- xy: 3, 10, 21, 36 — NOT constant.
This is a linear relationship (y = 2x + 1) but NOT direct proportion because the line does not pass through the origin.
Answer: Neither direct nor inverse proportion.
Real-World Applications
Real-world applications:
- Science experiments: Plotting graphs to discover whether two variables are directly or inversely proportional.
- Speed-distance-time problems: Distance vs time at constant speed gives a straight line (direct). Speed vs time for fixed distance gives a hyperbola (inverse).
- Economics: Supply-demand curves often resemble inverse proportion graphs.
- Engineering: Gear ratios and pulley systems involve inverse proportions.
- Cooking: Ingredients and servings are in direct proportion — the graph is a straight line.
Key Points to Remember
- Graph of direct proportion is a straight line through the origin.
- Graph of inverse proportion is a curve (hyperbola).
- For direct proportion, y/x = k (constant) and y = kx.
- For inverse proportion, xy = k (constant) and y = k/x.
- A straight line NOT through the origin is linear but not directly proportional.
- In direct proportion graph, slope = k (constant of proportionality).
- In inverse proportion graph, the curve never touches the axes.
- To identify the type, check ratios (y/x) for direct or products (xy) for inverse.
Practice Problems
- Plot the graph for y = 4x using x = 1, 2, 3, 4, 5. What type of proportion is this?
- Plot the graph for xy = 24. Use x = 1, 2, 3, 4, 6, 8, 12, 24.
- A car travels at 80 km/h. Plot distance vs time for t = 1, 2, 3, 4, 5 hours.
- If 240 km is covered at different speeds, plot speed vs time for speeds 30, 40, 60, 80.
- From this data — x: 3, 6, 9, 12 and y: 8, 16, 24, 32 — identify the proportion and describe the graph.
- Is the relation y = 5x + 2 a direct proportion? Why or why not?
Frequently Asked Questions
Q1. What does the graph of direct proportion look like?
It is a straight line that passes through the origin (0, 0). The steeper the line, the larger the constant of proportionality.
Q2. What does the graph of inverse proportion look like?
It is a curve called a hyperbola. It gets closer to the x-axis and y-axis but never touches them.
Q3. How do you identify the type of proportion from a graph?
Straight line through origin = direct proportion. Curve approaching axes = inverse proportion. Straight line not through origin = linear but not proportional.
Q4. Can the graph of direct proportion be a curve?
No. Direct proportion always gives a straight line. If the graph is curved, the quantities are not in direct proportion.
Q5. What is the slope of a direct proportion graph?
The slope equals the constant of proportionality k. If y = 3x, the slope is 3.
Q6. Does the inverse proportion graph pass through the origin?
No. Since y = k/x, when x = 0, y is undefined. The graph never touches or crosses the origin.
Q7. What is a hyperbola?
A hyperbola is a smooth curve formed when the product of two variables is constant (xy = k). It has two branches — for positive values, only the first quadrant branch is used.
Q8. How do you find k from a direct proportion graph?
Pick any point (x, y) on the line and calculate k = y/x. Alternatively, k equals the slope of the line.
