Direct Variation
When y is directly proportional to x, y=kx for constant k. Doubling x doubles y. Applies to Hooke's law, speed-distance-time, wages-hours. Power variation y=kxโฟ and joint variation y=kxz extend the concept.
Did our AI summary help? Let us know.
Direct variation graphs pass through the origin (0,0). Ratio y/x is constant for all (x,y) pairs on the line. Joint variation: y depends on the product of two variables.
Ready to run the numbers?
Why: Direct variation models real-world proportionality: spring force (Hooke), distance at constant speed, wages for hours worked. The graph is a line through the origin.
How: Given (x,y): k=y/x. Given k and x: y=kx. Given k and y: x=y/k. Power variation: k=y/xโฟ. Joint: k=y/(xz). The constant k is the slope of the line y=kx.
Run the calculator when you are ready.
๐ Examples โ Click to Load
Solve Mode
y Values for Different x
Constant k Contribution
๐ Calculation Steps
For educational and informational purposes only. Verify with a qualified professional.
๐งฎ Fascinating Math Facts
Hooke's law: F=kx โ force proportional to displacement.
โ Physics
d=rt: distance varies directly with time at constant speed.
โ Motion
๐ Key Takeaways
- โข Direct variation means y = kx: y is directly proportional to x with constant k
- โข The constant of variation k = y/x is the ratio of y to x โ it never changes along the line
- โข The graph of y = kx is a straight line through the origin (0, 0)
- โข Power variation y = kx^n extends this: e.g., area โ rยฒ, volume โ rยณ
- โข Joint variation y = kxz: one variable varies with the product of two others
๐ก Did You Know?
๐ How Direct Variation Works
In y = kx, k is the constant of proportionality. If you know one pair (x, y), you find k = y/x. Then you can predict any other y for a given x, or any x for a given y. The graph is always a line through the origin.
Example: y = 2x
If x = 3, then y = 6. The ratio 6/3 = 2 is constant. For x = 5, y = 10. The line passes through (0,0), (3,6), (5,10).
Graphing Through the Origin
Every direct variation y = kx passes through (0, 0). To graph: plot the origin, then use k as the slope โ for each 1 unit right, go k units up. The line extends in both directions.
๐ Table of Values (y = kx)
| x | y = 2x | y = -3x |
|---|---|---|
| -2 | -4 | 6 |
| -1 | -2 | 3 |
| 0 | 0 | 0 |
| 1 | 2 | -3 |
| 2 | 4 | -6 |
| 3 | 6 | -9 |
๐ Real-World Proportional Relationships
Many physical and economic relationships follow direct variation: distance = speed ร time, cost = price ร quantity, force = spring constant ร extension. When one variable doubles, the other doubles.
๐ฏ Expert Tips
๐ก Check the Origin
True direct variation always passes through (0,0). If your data doesn't, it may be linear but not direct (y = kx + b).
๐ก Ratio Test
For multiple points, compute y/x for each. If all ratios are equal, you have direct variation.
๐ก Power vs Linear
y = kxยฒ is not a line โ it's a parabola. Use power variation for area (โ rยฒ) or volume (โ rยณ).
๐ก Joint Variation
y = kxz: y varies with both x and z. Find k from one triple (x,y,z), then predict for other values.
๐ Reference: Variation Types
| Type | Formula |
|---|---|
| Direct | y = kx |
| Power | y = kx^n |
| Joint | y = kxz |
| Finding k | k = y/x (or y/(xz), y/x^n) |
โ FAQ
What is direct variation?
Direct variation means y = kx: one variable is a constant multiple of the other. The graph is a line through the origin.
What is the constant of variation?
The constant k in y = kx. It equals y/x for any point on the line. Also called the constant of proportionality.
How do I find k from two points?
Use either point: k = y/x. If both points lie on the same direct variation line, you get the same k from each.
What is power variation?
y = kx^n: y varies with the nth power of x. Examples: area โ rยฒ, volume โ rยณ.
What is joint variation?
y = kxz: y varies with the product of x and z. Example: force F = ma (mass ร acceleration).
Does direct variation always pass through the origin?
Yes. If y = kx, then when x = 0, y = 0. If your data has a non-zero y-intercept, it is linear but not direct variation.
๐ข Quick Reference
โ ๏ธ Disclaimer: This calculator assumes ideal direct, power, or joint variation. Real-world data may have measurement error or not follow these models exactly. Educational use only.
Related Calculators
Absolute Value Equation Calculator
Solve absolute value equations and inequalities including |ax+b|=c, |ax+b|=|cx+d|, and |ax+b|โคc forms. Features solution verification, number line...
MathematicsAbsolute Value Inequalities Calculator
Solve absolute value inequalities |ax+b| < c, > c, โค c, โฅ c. Get solution intervals, interval notation, set builder notation, number line visualization...
MathematicsBessel Function Calculator
Calculate Bessel functions J_n(x), Y_n(x), I_n(x), and K_n(x) with series approximation. Supports first kind (J), second kind/Neumann (Y), and modified Bessel functions (I, K). Applications include vibrating drum modes, electromagnetic waveguides, heat conduction in cylinders, and Bessel filter design in signal processing. Uses the power series J_n(x) = ฮฃ (-1)^m/(m!(m+n)!)ยท(x/2)^(2m+n). Includes convergence info, Bar chart comparing values at different x, Doughnut chart showing series term contributions, and educational content on Bessel's differential equation and cylindrical harmonics.
MathematicsBinomial Coefficient Calculator
Calculate binomial coefficients C(n,k) = n!/(k!(n-k)!) โ the number of ways to choose k items from n without regard to order. Also known as n choose k or โฟCโ. Features Pascal's triangle row generation, factorial breakdown, Bar chart of full row C(n,0)..C(n,n), Doughnut chart showing k vs nโk. Applications: lottery (C(49,6)), poker hands (C(52,5)), committee selection, DNA combinations, binary strings. Educational content on combinatorics, binomial theorem (x+y)โฟ expansion, and Pascal's identity.
MathematicsBox Method Calculator
Multiply polynomials visually using the box method (area model). Step-by-step solutions for binomials, trinomials, and factoring. Interactive grid...
MathematicsCompleting the Square Calculator
Convert quadratic expressions from standard form axยฒ+bx+c to vertex form a(x-h)ยฒ+k. Find vertex, axis of symmetry, x-intercepts, and min/max values....
Mathematics