🔵 Direct Proportion
\(y \propto x\)
Mathematical Form:
\[y = kx\]
where \(k\) is the constant of proportionality \((k > 0)\)
Meaning: When one quantity increases, the other increases proportionally. When one decreases, the other decreases proportionally.
📐 Mathematical Properties:
\[\frac{y_1}{x_1} = \frac{y_2}{x_2} = k \text{ (constant ratio)}\] \[\text{If } x \text{ becomes } nx, \text{ then } y \text{ becomes } ny\]Graph: Straight line through origin
Equation: \(y = 2x\)
Slope: \(m = k = 2\)
Numerical Example
| \(x\) | \(y = 2x\) | \(\frac{y}{x}\) |
|---|---|---|
| 1 | 2 | 2 |
| 2 | 4 | 2 |
| 3 | 6 | 2 |
| 4 | 8 | 2 |
| 5 | 10 | 2 |
🌟 Real-World Examples:
Speed & Distance:
At constant speed, distance is proportional to time
\[d = vt\]
where \(v\) is constant velocity
Cost & Quantity:
Total cost is proportional to number of items
\[\text{Cost} = \text{Price} \times \text{Quantity}\]
where price per item is constant
Hooke’s Law:
Force is proportional to extension
\[F = kx\]
where \(k\) is spring constant
Circumference:
Circumference is proportional to radius
\[C = 2\pi r\]
where \(k = 2\pi\)📝 Key Points:
- Graph passes through origin \((0,0)\)
- Slope \(= k\) (constant of proportionality)
- Ratio \(\frac{y}{x} = k\) remains constant
- Linear relationship: \(y = mx + c\) where \(c = 0\)
🔴 Inverse Proportion
\(y \propto \frac{1}{x}\)
Mathematical Form:
\[y = \frac{k}{x}\]
where \(k\) is the constant of proportionality \((k > 0)\)
Meaning: When one quantity increases, the other decreases proportionally. The product \(xy\) remains constant.
📐 Mathematical Properties:
\[x_1 y_1 = x_2 y_2 = k \text{ (constant product)}\] \[\text{If } x \text{ becomes } nx, \text{ then } y \text{ becomes } \frac{y}{n}\]Graph: Rectangular Hyperbola
Equation: \(y = \frac{12}{x}\)
Product: \(xy = 12\)
Numerical Example
| \(x\) | \(y = \frac{12}{x}\) | \(xy\) |
|---|---|---|
| 1 | 12 | 12 |
| 2 | 6 | 12 |
| 3 | 4 | 12 |
| 4 | 3 | 12 |
| 6 | 2 | 12 |
🌟 Real-World Examples:
Speed & Time:
Time is inversely proportional to speed (for fixed distance)
\[t = \frac{d}{v}\]
where \(d\) is constant distance
Workers & Time:
Time is inversely proportional to number of workers
\[t = \frac{W}{n}\]
where \(W\) is total work
Boyle’s Law:
Pressure is inversely proportional to volume
\[P = \frac{k}{V} \text{ or } PV = k\]
at constant temperature
Ohm’s Law:
Current is inversely proportional to resistance
\[I = \frac{V}{R}\]
at constant voltage \(V\)📝 Key Points:
- Graph is a rectangular hyperbola
- Never touches the axes (asymptotes)
- Product \(xy = k\) (constant)
- As \(x \to 0^+\), \(y \to +\infty\) and as \(x \to +\infty\), \(y \to 0^+\)
🟦 Direct Proportion to Square
\(y \propto x^2\)
Mathematical Form:
\[y = kx^2\]
where \(k\) is the constant of proportionality \((k > 0)\)
Meaning: The dependent variable is proportional to the square of the independent variable. Growth accelerates rapidly.
📐 Mathematical Properties:
\[\frac{y_1}{x_1^2} = \frac{y_2}{x_2^2} = k \text{ (constant ratio)}\] \[\text{If } x \text{ becomes } nx, \text{ then } y \text{ becomes } n^2y\]Graph: Parabola
Equation: \(y = 2x^2\)
Vertex: \((0,0)\)
Numerical Example
| \(x\) | \(y = 2x^2\) | \(\frac{y}{x^2}\) |
|---|---|---|
| 1 | 2 | 2 |
| 2 | 8 | 2 |
| 3 | 18 | 2 |
| 4 | 32 | 2 |
| 5 | 50 | 2 |
🌟 Real-World Examples:
Area of Square:
Area is proportional to square of side length
\[A = s^2\]
where \(k = 1\)
Kinetic Energy:
KE is proportional to square of velocity
\[KE = \frac{1}{2}mv^2\]
where \(k = \frac{m}{2}\)
Free Fall Distance:
Distance is proportional to square of time
\[s = \frac{1}{2}gt^2\]
where \(k = \frac{g}{2}\)
Area of Circle:
Area is proportional to square of radius
\[A = \pi r^2\]
where \(k = \pi\)📝 Key Points:
- Graph is a parabola opening upward
- Passes through origin \((0,0)\)
- Rate of increase accelerates with \(x\)
- Doubling \(x\) makes \(y\) four times larger
🟥 Inverse Proportion to Square
\(y \propto \frac{1}{x^2}\)
Mathematical Form:
\[y = \frac{k}{x^2}\]
where \(k\) is the constant of proportionality \((k > 0)\)
Meaning: The dependent variable is inversely proportional to the square of the independent variable. Decay is very rapid.
📐 Mathematical Properties:
\[x_1^2 y_1 = x_2^2 y_2 = k \text{ (constant product)}\] \[\text{If } x \text{ becomes } nx, \text{ then } y \text{ becomes } \frac{y}{n^2}\]Graph: Steep Hyperbola
Equation: \(y = \frac{36}{x^2}\)
Product: \(x^2y = 36\)
Numerical Example
| \(x\) | \(y = \frac{36}{x^2}\) | \(x^2y\) |
|---|---|---|
| 1 | 36 | 36 |
| 2 | 9 | 36 |
| 3 | 4 | 36 |
| 4 | 2.25 | 36 |
| 6 | 1 | 36 |
🌟 Real-World Examples:
Newton’s Law of Gravitation:
Gravitational force is inversely proportional to square of distance
\[F = \frac{GMm}{r^2}\]
where \(k = GMm\)
Coulomb’s Law:
Electric force is inversely proportional to square of distance
\[F = \frac{kq_1q_2}{r^2}\]
Electrostatic force
Light Intensity:
Intensity is inversely proportional to square of distance
\[I = \frac{P}{4\pi r^2}\]
Inverse square law
Electric Field:
Electric field strength varies as inverse square of distance
\[E = \frac{kQ}{r^2}\]
Point charge field📝 Key Points:
- Very steep hyperbolic curve
- Approaches zero extremely quickly as \(x\) increases
- Doubling \(x\) makes \(y\) one-fourth the original value
- Common in physics (inverse square laws)
🟧 Direct Proportion to Root
\(y \propto \sqrt{x}\)
Mathematical Form:
\[y = k\sqrt{x} = kx^{1/2}\]
where \(k\) is the constant of proportionality \((k > 0)\)
Meaning: The dependent variable is proportional to the square root of the independent variable. Growth rate decreases as \(x\) increases.
📐 Mathematical Properties:
\[\frac{y_1}{\sqrt{x_1}} = \frac{y_2}{\sqrt{x_2}} = k \text{ (constant ratio)}\] \[\text{If } x \text{ becomes } n^2x, \text{ then } y \text{ becomes } ny\]Graph: Square Root Curve
Equation: \(y = 3\sqrt{x}\)
Domain: \(x \geq 0\)
Numerical Example
| \(x\) | \(y = 3\sqrt{x}\) | \(\frac{y}{\sqrt{x}}\) |
|---|---|---|
| 1 | 3 | 3 |
| 4 | 6 | 3 |
| 9 | 9 | 3 |
| 16 | 12 | 3 |
| 25 | 15 | 3 |
🌟 Real-World Examples:
Simple Pendulum:
Time period is proportional to square root of length
\[T = 2\pi\sqrt{\frac{L}{g}}\]
where \(k = 2\pi\sqrt{\frac{1}{g}}\)
Escape Velocity:
Velocity is proportional to square root of mass
\[v = \sqrt{\frac{2GM}{r}}\]
For constant radius
RMS Velocity:
Velocity is proportional to square root of temperature
\[v_{rms} = \sqrt{\frac{3RT}{M}}\]
Kinetic theory of gases
Free Fall Time:
Time is proportional to square root of height
\[t = \sqrt{\frac{2h}{g}}\]
For free fall from rest📝 Key Points:
- Starts at origin, increases at decreasing rate
- Concave downward curve (decreasing slope)
- Domain restricted to \(x \geq 0\) for real values
- Common in physics formulas involving square roots
🟨 Inverse Proportion to Root
\(y \propto \frac{1}{\sqrt{x}}\)
Mathematical Form:
\[y = \frac{k}{\sqrt{x}} = kx^{-1/2}\]
where \(k\) is the constant of proportionality \((k > 0)\)
Meaning: The dependent variable is inversely proportional to the square root of the independent variable. Moderate decay rate.
📐 Mathematical Properties:
\[y_1\sqrt{x_1} = y_2\sqrt{x_2} = k \text{ (constant product)}\] \[\text{If } x \text{ becomes } n^2x, \text{ then } y \text{ becomes } \frac{y}{n}\]Graph: Moderate Decay
Equation: \(y = \frac{12}{\sqrt{x}}\)
Product: \(y\sqrt{x} = 12\)
Numerical Example
| \(x\) | \(y = \frac{12}{\sqrt{x}}\) | \(y\sqrt{x}\) |
|---|---|---|
| 1 | 12 | 12 |
| 4 | 6 | 12 |
| 9 | 4 | 12 |
| 16 | 3 | 12 |
| 25 | 2.4 | 12 |
🌟 Real-World Examples:
Diffusion Process:
Concentration is inversely proportional to square root of time
\[C = \frac{C_0}{\sqrt{t}}\]
In some diffusion models
Wave Amplitude:
Amplitude decreases with square root of distance
\[A = \frac{A_0}{\sqrt{r}}\]
Cylindrical wave propagation
Frequency Relationship:
In some systems, frequency varies inversely with square root
\[f = \frac{k}{\sqrt{L}}\]
String vibrations
Statistical Distribution:
Standard error inversely proportional to square root of sample size
\[SE = \frac{\sigma}{\sqrt{n}}\]
Central limit theorem📝 Key Points:
- Decays slower than \(\frac{1}{x^2}\) but faster than \(\frac{1}{x}\)
- Approaches zero but never reaches it
- Domain: \(x > 0\)
- Vertical asymptote at \(x = 0\), horizontal asymptote at \(y = 0\)
📋 Complete Mathematical Summary
| Type | Formula | Constant Property | Graph Shape | Growth/Decay Rate |
|---|---|---|---|---|
| Direct | \(y = kx\) | \(\frac{y}{x} = k\) | Straight line | Constant rate |
| Inverse | \(y = \frac{k}{x}\) | \(xy = k\) | Hyperbola | Moderate decay |
| Direct Square | \(y = kx^2\) | \(\frac{y}{x^2} = k\) | Parabola | Accelerating growth |
| Inverse Square | \(y = \frac{k}{x^2}\) | \(x^2y = k\) | Steep hyperbola | Rapid decay |
| Direct Root | \(y = k\sqrt{x}\) | \(\frac{y}{\sqrt{x}} = k\) | Root curve | Slowing growth |
| Inverse Root | \(y = \frac{k}{\sqrt{x}}\) | \(y\sqrt{x} = k\) | Moderate decay | Medium decay |