Quadratic Functions and Equations in One Variable · 1.1.2
Quadratic functions as relations
This standard builds on relations: a quadratic function f(x) = ax² + bx + c, a ≠ 0, maps two different x-values to the same y-value, except at the turning point, making it many-to-one. Students use this to describe features such as the parabola shape, turning point, and axis of symmetry.
The official learning standard (1.1.2)
“Recognise quadratic function as a many-to-one relation, hence describe the characteristics of quadratic functions.”
What it means
This standard builds on relations: a quadratic function f(x) = ax² + bx + c, a ≠ 0, maps two different x-values to the same y-value, except at the turning point, making it many-to-one. Students use this to describe features such as the parabola shape, turning point, and axis of symmetry.
How it is examined
Paper 1 often tests this through identifying many-to-one relations from graphs, arrow diagrams, or ordered pairs, and matching them to quadratic functions. It also appears as background understanding for sketching and interpreting quadratic graphs in Paper 2.
Worked example
Given the quadratic function f(x) = x² − 4, show that it is a many-to-one relation by finding f(3) and f(−3), then state one characteristic of its graph.
- f(3) = (3)² − 4 = 9 − 4 = 5
- f(−3) = (−3)² − 4 = 9 − 4 = 5
- Two different x-values, 3 and −3, both map to the same y-value, 5, so f is a many-to-one relation.
- The graph is a parabola with a minimum turning point at (0, −4) and axis of symmetry x = 0.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What does "many-to-one" mean for a quadratic function?
It means two (or more) different x-values can produce the same y-value. For f(x) = x², both f(1) and f(−1) equal 1, so the inputs 1 and −1 are mapped to the same output.
How is this different from a one-to-one relation like a linear function?
In a one-to-one linear function, each x-value gives a unique y-value, and no two x-values share the same output. A quadratic function pairs up x-values symmetrically about its turning point, so the mapping is many-to-one instead.
How can I identify a many-to-one relation from a graph?
Draw a horizontal line across the graph. If it crosses the curve at two points (except at the turning point, where it touches once), the relation is many-to-one, this matches the shape of a parabola.