Direct and Inverse Variation
Direct: y = kx. Inverse: y = k/x.
About this topic
In direct variation, y = kx, the ratio y/x stays constant, so doubling x doubles y; the graph is a line through the origin. In inverse variation, y = k/x, the product xy is constant, so doubling x halves y, giving a hyperbola.
Find k from one known pair of values, then use it for any other. Direct variation describes wages for hours worked, while inverse variation describes travel time versus speed or the pressure of a gas versus its volume.
Worked example: if y varies directly with x and y = 12 when x = 4, then k = 3, so y = 3x and x = 10 gives y = 30. If instead y varies inversely and y = 12 when x = 4, then k = 48 and x = 10 gives y = 4.8. A car at 60 km per hour covers 120 km in 2 hours, but at 40 km per hour it needs 3 hours: speed and time vary inversely with constant product, the distance.
The inverse model also explains why a pizza shared among more people gives each a smaller slice. Students sometimes call any relationship where one quantity goes up while another goes down inverse variation; it must follow xy = k exactly. Check by multiplying several pairs. For direct variation, check the ratios y/x. A line that does not pass through the origin, such as y = 2x + 5, is not direct variation.
Key ideas
- Direct variation gives a line through the origin
- Inverse variation gives a hyperbola
- Product xy is constant for inverse
Quick check
Use cases: where Direct and Inverse Variation helps
- Homework: review the key idea behind Direct and Inverse Variation before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Direct and Inverse Variation without notes.
- Visual learning: play with the interactive lab to see how Direct and Inverse Variation changes when you adjust its inputs.
- Teaching a friend: explain Direct and Inverse Variation in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Direct and Inverse Variation?
Direct: y = kx. Inverse: y = k/x.
What will I learn in Direct and Inverse Variation?
Direct variation gives a line through the origin. Inverse variation gives a hyperbola. Product xy is constant for inverse.
Is Direct and Inverse Variation part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Direct and Inverse Variation lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Direct and Inverse Variation?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.