y varies inversely as the square of x, and y = 5 when x = 4. Find y when x = 10.
- Write the model: y = k/x2.
- Find k: 5 = k/42, so k = 5 × 16 = 80.
- When x = 10: y = 80/102 = 80 ÷ 100 = 0.8.
Answer: y = 0.8
Linear, quadratic, cubic and variation models
Maths: Applications and Interpretation · Functions. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
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The idea
A model is a function chosen to describe a real situation. A linear model has a constant rate of change. A piecewise linear model uses different straight lines on different parts of the domain, such as a fixed charge followed by a charge per kilometre.
A quadratic model has one turning point, the vertex, which gives the maximum or minimum value. A cubic model can have two turning points, and you find its maximum on a domain with your calculator.
Variation models have the form f(x) = axn. Direct variation has a positive power. Inverse variation has a negative power, so one quantity falls as the other rises. Find the constant first, then use it.
Rules to remember
- Linear model: f(x) = mx + c, with rate of change m and starting value c
- Piecewise model: first decide which piece the x value belongs to, then use only that rule
- Quadratic f(x) = ax2 + bx + c: axis of symmetry x = −b/(2a); the vertex is a maximum if a < 0 and a minimum if a > 0
- The axis of symmetry is halfway between the two zeros, or between any two points with the same y value
- y varies directly as xn: y = kxn
- y varies inversely as xn: y = k/xn, so y × xn is constant
- The domain of a model is limited by the context, e.g. lengths and times cannot be negative
Common mistake
In variation questions students scale the answer in simple proportion and ignore the power. If y varies as the square of x, doubling x multiplies y by 4. Find k from the given pair of values first.
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