More about graphs of functions
How to solve equations and inequalities from graphs, and how graphs move under translations, reflections and enlargements. The exponential and logarithmic graphs and the transformations are NF.
What you will learn
- 9.1 Sketch and compare graphs: constant, linear, quadratic, trigonometric, and NF exponential and logarithmic. Compare domains, maximum and minimum, symmetry, periodicity.
- 9.2 Solve f(x) = k from the graph of y = f(x).
- 9.3 Solve f(x) > k, f(x) < k, f(x) ≥ k, f(x) ≤ k from the graph.
- NF 9.4 Transformations: f(x) + k, f(x + k), kf(x), f(kx), from tables, symbols and graphs.
You need first: Functions and graphs, Trigonometry (2-D), Exponential and logarithmic functions
Key terms
| English | 中文 |
|---|---|
| translation | 平移 |
| reflection | 反射 |
| enlargement / reduction (stretch) | 放大／縮小（伸展） |
| along the x-axis / y-axis | 沿 x 軸／y 軸 |
| transformation of graphs | 函數圖像的變換 |
| periodicity | 週期性 |
Standard steps
Solve f(x) = k or f(x) > k from a graph
- Draw the horizontal line y = k.
- f(x) = k: read the x-values where the curve meets the line.
- f(x) > k: read the x-values where the curve is above the line.
- To solve f(x) = g(x), read where the two graphs cross.
Transformations (NF): "outside acts on y, inside acts on x, and inside does the opposite"
| New graph | What happens | Point (a, b) becomes |
|---|---|---|
| y = f(x) + k | up k units | (a, b + k) |
| y = f(x + k) | left k units | (a − k, b) |
| y = kf(x) | stretch along y-axis by k | (a, kb) |
| y = f(kx) | squeeze along x-axis by k (x ÷ k) | (a/k, b) |
| y = −f(x) | reflect in the x-axis | (a, −b) |
| y = f(−x) | reflect in the y-axis | (−a, b) |
Check with one point: the vertex of a parabola is the easiest.
Worked examples
The parabola y = x² − 2x − 3 and the line y = x + 1 meet where x² − 3x − 4 = 0, so at x = −1 and x = 4. Reading the graph should give the same values.
"left 3" is f(x + 3): y = (x + 3)² + 1
vertex (0, 1) → (−3, 1)Right 2 and up 3: (4, 8).
Common mistakes
- f(x + 3) moved to the right (it moves left).
- Mixing up 2f(x) (taller) and f(2x) (narrower).
- Solving f(x) > k by reading where the curve is below the line.
- Reading graph values too roughly; use the grid and the question's accuracy.
How the DSE tests it
- Paper 2 Section A: read solutions of f(x) = k or f(x) ≥ k from a given graph.
- Paper 2 Section B: identify a transformed graph or the transformation from two equations.
- Section B long question: complete the square, then describe a translation.
Questions you might not ask in class
Why does f(x + 3) move left, not right?
The new graph reaches each height 3 units earlier. At x = −3 it already gives f(0), so everything happens 3 units further left.
Can I just use the calculator to solve instead of the graph?
Algebra is fine for checking, but "use the graph" questions want values read from the graph, and the marks are for reading it.