1.2 Inverse Variation
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.
Learning Objectives
- Explain the meaning of inverse variation.
- Determine the relationship between two variables for an inverse variation.
- Express an inverse variation in the form of a variation relation and an equation.
- Represent inverse variations graphically and determine the constant of variation from the gradient of a straight-line graph.
- Solve problems involving inverse variation in real-world contexts.
Comparison of Graphs in 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.
Brief Explanation
What is Inverse Variation?
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".
General Form and Formulae
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 \(n = 1, 2, 3, \frac{1}{2}, \frac{1}{3}\).
| Specific form | Variation relation | Equation form |
|---|---|---|
| \(y\) varies inversely as \(x\) | \(y \propto \dfrac{1}{x}\) | \(y = \dfrac{k}{x}\) |
| \(y\) varies inversely as the square of \(x\) | \(y \propto \dfrac{1}{x^2}\) | \(y = \dfrac{k}{x^2}\) |
| \(y\) varies inversely as the cube of \(x\) | \(y \propto \dfrac{1}{x^3}\) | \(y = \dfrac{k}{x^3}\) |
| \(y\) varies inversely as the square root of \(x\) | \(y \propto \dfrac{1}{\sqrt{x}}\) | \(y = \dfrac{k}{\sqrt{x}}\) |
| \(y\) varies inversely as the cube root of \(x\) | \(y \propto \dfrac{1}{\sqrt[3]{x}}\) | \(y = \dfrac{k}{\sqrt[3]{x}}\) |
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
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.
Common Mistakes
- Confusing direct and inverse variation equations: writing \(y = kx\) instead of \(y = \frac{k}{x}\) when setting up the relation.
- Misinterpreting the gradient: thinking the gradient of the graph of \(y\) against \(x\) gives \(k\), when only the gradient of \(y\) against \(\frac{1}{x^n}\) gives \(k\).
- Forgetting powers or roots: when a question states “inversely as the square of \(x\)” or “square root of \(x\)”, students forget to square or square-root the value when calculating \(k\).
- Incorrect proportion setup: writing \(\frac{y_1}{x_1} = \frac{y_2}{x_2}\) instead of \(x_1 y_1 = x_2 y_2\) or \(\frac{y_1}{y_2} = \frac{x_2}{x_1}\).
Misconceptions
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.
Summary
- Inverse variation means that as one variable increases, the other decreases at the same rate.
- Algebraic relation: \(y \propto \dfrac{1}{x^n}\) is written as an equation \(y = \dfrac{k}{x^n}\), which means \(x^n y = k\).
- The graph of \(y\) against \(x\) is a hyperbola.
- The graph of \(y\) against \(\dfrac{1}{x^n}\) is a straight line that starts from the origin with gradient \(k\).
- Unknown values can be found using the constant equation \(y = \dfrac{k}{x^n}\) or the proportion method \(x_1^{\,n} y_1 = x_2^{\,n} y_2\).
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
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 \(t\)?
B. Fill in the Blanks
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 .
C. Drag and Drop: Match Variation to Equation
Drag each description to the correct equation form. On a phone, tap the answer choice first, then tap the equation box.
