1.1 Direct Variation
Direct variation explains the relationship between two variables such that when variable $y$ increases, variable $x$ also increases at the same rate, and vice versa.
Mathematical and Relation Representations
- Variation relation: $y \propto x^n$
- Equation form: $y = k x^n$, where $k$ is a constant and $k \neq 0$.
Common powers of $n$ in Form 5 KSSM include $n = 1, 2, 3, \frac{1}{2}, \frac{1}{3}$.
Graphical Representation
- For $y \propto x$, the graph of $y$ against $x$ is a straight line that passes through the origin $(0,0)$.
- The gradient of the straight line represents the constant of variation, $k$.
1.2 Inverse Variation
Inverse variation explains the relationship between two variables such that when variable $y$ increases, variable $x$ decreases at the same rate, and vice versa.
Mathematical and Relation Representations
- Variation relation: $y \propto \frac{1}{x^n}$
- Equation form: $y = \frac{k}{x^n}$, where $k$ is a constant and $k \neq 0$.
Graphical Representation
- The graph of $y$ against $\frac{1}{x^n}$ is a straight line passing through the origin $(0,0)$ with gradient $k$.
- The graph of $y$ against $x$ forms a hyperbola curve that does not touch the axes.
1.3 Joint Variation
Joint variation is a variation where a variable varies as a product of two or more variables (can be a combination of direct and/or inverse variations).
Key Formulations
- Joint Direct Variation: If $y$ varies directly as $x$ and $z$, then $y \propto xz \implies y = k x z$.
- Combined Variation (Direct & Inverse): If $y$ varies directly as $x^m$ and inversely as $z^n$, then:
$$\text{Relation: } y \propto \frac{x^m}{z^n} \implies \text{Equation: } y = \frac{k x^m}{z^n}$$
Summary of Steps to Solve Variation Problems
- Write the relation using the symbol $\propto$.
- Express the relation in equation form with the constant $k$.
- Substitute the given values of variables into the equation to find the value of $k$.
- Rewrite the equation with the determined numerical value of $k$.
- Substitute remaining given values to solve for the unknown variable.