Functions and graphs
Functions and f(x) notation, the graph of a quadratic function (vertex, axis of symmetry, intercepts), and (NF) finding a maximum or minimum by completing the square. It is the base of later graph questions in DSE Paper 1.
What you will learn
- 2.1 Functions; domain and co-domain; independent and dependent variables.
- 2.2 f(x) notation; tables, algebra and graphs to represent functions.
- 2.3 Graph of a quadratic function: vertex, axis of symmetry, direction of opening, intercepts; maximum or minimum value from the graph.
- NF 2.4 Maximum and minimum by completing the square; problems about maximum and minimum.
You need first: Quadratic equations in one unknown, Coordinates
Key terms
| English | 中文 |
|---|---|
| function | 函數 |
| domain, co-domain | 定義域，上域 |
| independent / dependent variable | 自變量／應變量 |
| quadratic function | 二次函數 |
| parabola | 拋物線 |
| vertex | 頂點 |
| axis of symmetry | 對稱軸 |
| opens upwards / downwards | 向上張開／向下張開 |
| maximum / minimum value | 極大值／極小值 |
| completing the square | 配方法 |
Standard steps
f(x) notation
f(a) means "put a in place of x". Always use brackets: f(−1) = (−1)² − 6(−1) + 5.
Sketch y = ax² + bx + c
- Direction: a > 0 opens up (has a minimum); a < 0 opens down (has a maximum).
- y-intercept: c.
- x-intercepts: solve ax² + bx + c = 0 (if Δ ≥ 0).
- Axis of symmetry: x = −b/(2a), or halfway between the x-intercepts.
- Vertex: put that x into the function.
Completing the square (NF)
Write y = a(x − h)² + k. The vertex is (h, k).
- Take out a from the x² and x terms.
- Inside the bracket, add and subtract (half the x-coefficient)².
- Tidy into a(x − h)² + k.
Why: (x − h)² is never negative, so its smallest value is 0, at x = h.
Worked examples
Example 1: Sketch y = f(x) for f(x) = x² − 6x + 5: find f(−1), the intercepts and the vertex.
f(−1) = 1 + 6 + 5 = 12
y-intercept: 5
x-intercepts: (x − 1)(x − 5) = 0, so x = 1, 5
axis: x = 3 (halfway between 1 and 5)
vertex: f(3) = 9 − 18 + 5 = −4, so (3, −4)Example 2: NF: find the maximum value of y = −2x² + 8x − 3 by completing the square.
y = −2(x² − 4x) − 3
= −2(x² − 4x + 4 − 4) − 3
= −2(x − 2)² + 8 − 3
= −2(x − 2)² + 5
maximum value 5, when x = 2Common mistakes
- f(−1) without brackets: −1² is −1, not 1.
- Vertex x-coordinate sign: (x − 3)² has vertex at x = +3.
- Completing the square with a ≠ 1, and forgetting to multiply the subtracted number by a: −2(−4) = +8.
- Calling the minimum point "the minimum value". The value is the y-coordinate only.
How the DSE tests it
- Section A: read the vertex, axis or intercepts from a graph or equation; Paper 2 matches graphs to equations (signs of a, b, c).
- Section B: completing the square in part (a), then a translated graph or a geometric property in later parts.
- Word problems: maximum area or maximum profit.
Questions you might not ask in class
What is a function, really?
A machine: one input goes in, exactly one output comes out.
Why is the vertex of (x − 3)² at +3, not −3?
The bracket is 0 when x = 3. That's where the lowest point is.
Why do we add and subtract the same number?
Adding 4 and taking away 4 changes nothing, but it lets us group x² − 4x + 4 into a perfect square.