Interpreting Proportion Equations
Interpreting Proportion Equations
Proportion relationships are expressed using the proportionality symbol \( \propto \) or as equations of the form \( y = kx^n \) or \( y = k/x^n \). Being able to read and interpret these equations — identifying what type of proportion exists, what the constant means, and what the graph looks like — is the key skill at this level.
Reading the Equation
| Equation | Proportion type | Graph shape | Key test |
|---|---|---|---|
| \( y = kx \) | Direct (linear) | Straight line through origin | \( y/x \) constant |
| \( y = kx^2 \) | Direct (quadratic) | Parabola through origin | \( y/x^2 \) constant |
| \( y = k\sqrt{x} \) | Direct (square root) | Half-parabola through origin | \( y/\sqrt{x} \) constant |
| \( y = k/x \) | Inverse | Hyperbola | \( xy \) constant |
| \( y = k/x^2 \) | Inverse square | Steeper hyperbola | \( x^2 y \) constant |
Interpreting in Context
When given an equation in a context, interpret \( k \) as the constant of proportionality — it has units that make the equation dimensionally consistent. For example, in \( d = 5t^2 \), if \( d \) is in metres and \( t \) in seconds, then \( k = 5 \) has units m/s².
Worked Examples
State the type of proportion and describe the graph for each: (a) \( F = 6r^2 \) (b) \( I = 120/d^2 \) (c) \( T = 3\sqrt{l} \)
(a) \( F \propto r^2 \): direct (quadratic). Parabola through origin.
(b) \( I \propto 1/d^2 \): inverse square. Steeply decreasing curve, approaching both axes.
(c) \( T \propto \sqrt{l} \): direct (square root). Increasing curve through origin, concave downward.
A table shows \( x \): 1, 4, 9, 16 and \( y \): 3, 6, 9, 12. Identify the proportion type and find the equation.
Check \( y/\sqrt{x} \): \( 3/1=3 \), \( 6/2=3 \), \( 9/3=3 \), \( 12/4=3 \) — constant. So \( y \propto \sqrt{x} \).
\[ y = 3\sqrt{x} \]The stopping distance \( d \) metres of a car is proportional to the square of its speed \( v \) km/h. At 40 km/h, the stopping distance is 24 m. Find \( d \) at 60 km/h.
\[ d = kv^2 \qquad k = 24/1600 = 0.015 \] \[ d = 0.015 \times 3600 = 54 \text{ m} \]Key Takeaways
- Read the equation to identify the power of \( x \) — this tells you the proportion type and the shape of the graph.
- To identify type from a table: test \( y/x \), then \( y/x^2 \), then \( y/\sqrt{x} \), then \( xy \) — whichever is constant gives the type.
- The constant \( k \) has units — always state what it means in context.
- Once \( k \) is found from one pair, it applies to all pairs in the same proportional relationship.