Functions and Graphs
Master quadratic, exponential, and logarithmic functions. Learn the exact rules of graph transformations and conquer the trickiest Paper 2 graph-matching questions.
Click the graph to explore it interactively →
The Core Concept
How to Read a Graph: Every function equation y = f(x) represents a geometric shape. Changing the numbers inside the equation transforms the shape (shifting, reflecting, or stretching it). In the DSE, you must be able to look at an equation and instantly visualize its basic shape, intercepts, and boundaries (asymptotes).
DSE Functions and Graphs Cheat Sheet — 5 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. Quadratic | Vertex Form:y = a(x − h)² + k• Vertex: (h, k)• Axis of Symmetry: x = h• Y-intercept: Let x = 0 | If a > 0 ⟹ Opens upward (minimum value = k).If a < 0 ⟹ Opens downward (maximum value = k). |
| 2. Exponential | y = aˣ(where a > 0 and a ≠ 1)• Passes through (0, 1).• Horizontal Asymptote: y = 0• Always positive ( y > 0). | Increasing if a > 1.Decreasing if 0 < a < 1. |
| 3. Logarithmic | y = logₐ x(where x > 0 and a ≠ 1)• Passes through (1, 0).• Vertical Asymptote: x = 0• Domain: x > 0. | Inverse of y = aˣ.The graphs are reflections of each other along the line y = x. |
| 4. Graph Translations | • Vertical: y = f(x) ± k• Horizontal: y = f(x ∓ h) | • +k shifts UP; −k shifts DOWN.• −h shifts RIGHT; +h shifts LEFT.Be extremely careful with horizontal shifts: f(x − 3) shifts to the right by 3 units. |
| 5. Reflections & Stretches | • Reflection:−f(x) or f(−x)• Stretch: a · f(x) or f(kx) | • −f(x): Reflect across X-axis.• f(−x): Reflect across Y-axis.• a · f(x): Vertical stretch by factor a.For y = f(kx), the graph is compressed horizontally by a factor of 1/k (not stretched by k). |
💡 Exam Traps & Secrets
⚠️ Horizontal Translation Direction TrapThe direction of horizontal translations is the opposite of what most students intuitively think.
Remember: "Minus goes Right, Plus goes Left" when modifying the x inside the parenthesis.
y = f(x − 4) ⟹ Shifts RIGHT by 4 units.y = f(x + 4) ⟹ Shifts LEFT by 4 units.Remember: "Minus goes Right, Plus goes Left" when modifying the x inside the parenthesis.
⚠️ The Log Domain Trap (Paper 1)You can never take the logarithm of a negative number or zero. In inequality or domain questions, the expression inside the logarithm must be strictly positive: For
y = logₐ(g(x)), you must set the domain condition to: g(x) > 0.⚠️ Paper 2 Graph-Matching Speed HackIn Paper 2, you are often asked to match a complex equation to one of four given graph sketches. Do not try to plot the whole function. Instead, test extreme values or key points:
• Let
• Check what happens to y as
• Draw a vertical line at
• Let
x = 0 to find the y-intercept.• Check what happens to y as
x → ∞ or x → −∞.• Draw a vertical line at
x = 1 on exponential graphs to easily compare which base is larger (y = aˣ vs. y = bˣ).✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Completing the Square)Express
y = −2x² + 12x − 13 in vertex form and state its vertex, axis of symmetry, and maximum value.(Exponential Graph Features)Find the horizontal asymptote and the y-intercept of the function
y = 3ˣ − 4.(Combined Horizontal Compression and Vertical Translation)The graph of
y = f(x) is given. Sketch the transformed graph of y = f(2x) − 1 on the same axes.Functions and Graphs Exercises: DSE-Style Practice
DSE-style practice sets from the SketchMath question bank, with solutions for every MC and long question.
Functions and Graphs · Form 4
34 MC questions · 4 long questions
Frequently Asked Questions
Complete the square to write y = a(x − h)² + k. The vertex is (h, k). Alternatively, use x = −b/(2a) for the x-coordinate, then substitute to find y. If a > 0, the vertex is a minimum; if a < 0, it is a maximum.
DSE tests translations (±k on y, ±h on x — where "minus goes right, plus goes left"), reflections (y = −f(x) across x-axis, y = f(−x) across y-axis), and vertical stretches (y = af(x)). Focus on how each moves key points: vertex, intercepts, asymptote.
Multiple-choice items frequently ask you to identify the graph of a transformed function, or match a stated property (e.g., "has a minimum and two positive x-intercepts") to the correct equation. Speed hack: test x = 0 for the y-intercept, and check asymptotic behavior as x → ±∞, rather than trying to plot the entire function.