Functions and Graphs
This domain concerns proportional and inverse-proportional relationships, linear functions, and quadratic functions, together with their graphs. You must connect an equation, a table, and a graph, and read features such as slope and intercept from any of them.
Core skills
- Recognise direct proportion y=ax (a straight line through the origin) and inverse proportion y=a/x (a curve with two branches).
- For a linear function y=ax+b, read the slope a as rise over run and b as the y-intercept.
- For a quadratic y=ax², know that a>0 opens upward and a<0 opens downward, and that the vertex is at the origin for this form.
- Determine a function's equation from given points by substituting and solving for the constant.
Common traps
Confusing direct and inverse proportion is common: in inverse proportion, as x doubles, y halves, so the product xy stays constant. Students also misread slope sign when the line falls from left to right (negative slope). When a domain is restricted, the graph is only a segment or ray, not the whole line. Do not assume a graph passes through the origin without checking.
Worked example
A quantity y is inversely proportional to x, and y=6 when x=2. Find y when x=3. The constant is xy=2×6=12, so y=12/x. When x=3, y=12/3=4.
Rule of thumb: identify the type of relationship first, find the constant from one known pair, then use it for the unknown.