Math101 · Independent Ontario learning library
Math101learn.math101.caVariation and Modeling
Variation models express how quantities change together through direct, inverse, joint, and power relationships.
A variation statement identifies the shape of a relationship; data or one known case determines its constant.
Direct variation
If $y$ varies directly with $x$, then
where $k$ is the constant of variation. For nonzero $x$,
The graph is a line through the origin. Doubling $x$ doubles $y$.
Partial variation
A relationship
with $b\ne0$ has a constant additive starting value plus direct change. It is linear but not directly proportional.
A taxi fare with a base charge and per-kilometre rate is a common partial-variation model.
Inverse variation
If $y$ varies inversely with $x$, then
so
Doubling $x$ halves $y$. The graph is a reciprocal-type curve rather than a line.
Worked example: inverse model
The constant $240$ represents the trip distance in kilometres.
Joint variation
If $y$ varies jointly with $x$ and $z$,
For a fixed $z$, $y$ varies directly with $x$, and vice versa. Geometry formulas often show joint variation: rectangle area $A=lw$ has $k=1$.
Combined variation
A quantity may vary directly with some variables and inversely with others:
Translate words carefully. “Varies directly as the square of $x$ and inversely as $z$” determines the powers and placement.
Power models
More generally,
The exponent $n$ describes scaling. If $n=2$, doubling $x$ multiplies $y$ by $4$; if $n=3$, by $8$; if $n=-1$, the model is inverse variation.
Area and volume scaling are power relationships.
Determining a model
- Translate the variation statement into an equation with $k$.
- substitute a known data point to solve for $k$.
- write the complete model with units/domain.
- use it to predict or solve.
- check the prediction against the relationship's direction.
Do not substitute the target case before the constant is known.
Identifying models from data
Direct variation has constant $y/x$; inverse variation has constant product $xy$. A power model can be investigated through ratios, log plots, or regression.
Real data will rarely produce perfectly constant values, so compare residuals and context rather than forcing exactness.
Model assumptions and limits
A variation model simplifies reality. Fixed travel distance may be clear, but speed might not remain constant. Area formulas assume ideal shapes. Financial or physical relationships may change outside observed ranges.
State domain, units, and whether a result is interpolation or extrapolation.
Common mistakes
Treating every linear equation as direct variation. Direct variation requires zero intercept.
Writing inverse variation as $y=k-x$. It is multiplicative: $y=k/x$.
Losing exponents in verbal translation. “Square of” matters.
Using a different $k$ for each case. The constant belongs to one model under stable conditions.
Ignoring units and realistic domain. A formula may be mathematically valid where the context is not.
Quick self-check
- Is the relationship direct, partial, inverse, joint, combined, or power variation?
- What equation translates the wording?
- Which known case determines $k$?
- Do units of $k$ make sense?
- Does the prediction move in the direction the relationship suggests?
- Is the model being used only over a defensible domain?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Travel time varies inversely with speed for a fixed trip. A trip takes 3 h at 80 km/h. How long at 100 km/h?
- k = 3 × 80 = 240.
- t = 240/100 = 2.4 hours.
End of lesson
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