Inverse variation describes a mathematical relationship between two variables where an increase in one variable causes a decrease in the other variable at the same rate, and vice versa. For instance, on a balanced seesaw, a heavier person must sit closer to the fulcrum while a lighter person sits further away. Similarly, as the number of open doors in a hall increases, the time taken for pupils to leave decreases. This topic covers the definition, algebraic expressions, graphical representations, and real-life applications of inverse variation.
Distinguish between the non-linear hyperbolic graph of \(y\) against \(x\) and the linear graph of \(y\) against \(\frac{1}{x}\) with gradient \(k\).
A smooth curved downward line known as a hyperbola. It never touches the axes.
A straight line starting from the origin. The gradient represents the constant of variation, k.
In an inverse variation, variable \(y\) increases when variable \(x\) decreases at the same rate, and vice versa. The relation can be written as "\(y\) varies inversely as \(x\)" or "\(x\) and \(y\) are inversely proportional".
Variation relation: \(y \propto \dfrac{1}{x^n}\)Equation form: \(y = \dfrac{k}{x^n}\) or \(x^n y = k\), where \(k\) is a non-zero constant and \(n = 1, 2, 3, \frac{1}{2}, \frac{1}{3}\).
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It is given that \(y\) varies inversely as \(x\), and \(y = 4\) when \(x = 3\). Express \(y\) in terms of \(x\), and determine the value of \(y\) when \(x = 6\).
The table shows values of two variables, \(p\) and \(q\): \(p\) = 2, 4, 5 and \(q\) = 18, 9, 7.2. Show whether \(q\) varies inversely as \(p\).
The force of gravity, \(F\), varies inversely as the square of the distance between two objects, \(d\). When \(d = 2\) m, \(F = 20\) N. Calculate \(F\) when \(d = 4\) m.
Inverse variation simply means when one variable increases, the other decreases.
While one increases as the other decreases, they must do so at the same rate such that their product \(x^n y\) is strictly a constant \(k\).
When \(y = \frac{1}{x^2}\), \(y\) varies directly as \(x\).
\(y\) varies directly as \(\frac{1}{x^2}\), but \(y\) varies inversely as \(x^2\). It does not vary directly as \(x\).
The graph of \(y\) against \(x\) in an inverse variation passes through the origin.
The graph of \(y\) against \(x\) is a hyperbola that never touches or intersects the axes. Only the graph of \(y\) against \(\frac{1}{x^n}\) is a straight line extending toward the origin.
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1 Which of the following describes the shape of the graph of \(y\) against \(x\) when \(y\) varies inversely as \(x\)?
2 It is given that \(t\) varies inversely as \(x\). If \(x\) is multiplied by 2, what happens to the value of \(t\)?
3 For an inverse variation where \(y\) varies inversely as \(x^n\), the gradient of the straight-line graph of \(y\) against \(\frac{1}{x^n}\) represents the of variation.
4 If \(y\) varies inversely as the square root of \(x\), and \(y = 3\) when \(x = 0.25\), the constant \(k\) is equal to .
Drag each description to the correct equation form. On a phone, tap the answer choice first, then tap the equation box.
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