Functions: The Logic of Transformation

#

In Grade 10, you sketched $y = ax^2 + q$, $y = \frac{a}{x} + q$, and $y = ab^x + q$ — graphs that only shifted up and down. In Grade 11, we add the horizontal shift ($p$), which moves every graph left or right. This one extra parameter changes everything: the turning point moves, the asymptotes move, and the methods for finding equations get more powerful.

The Universal Parameter Table

#

Every Grade 11 function follows this pattern. Memorise what each parameter does:

⚠️ The $p$-sign trap: In $y = a(x - p)^2 + q$, if the bracket says $(x - 3)$, then $p = 3$ (shift RIGHT). If it says $(x + 2)$, then $p = -2$ (shift LEFT). The sign inside the bracket is opposite to the direction of the shift.

The Three Functions at a Glance

#

What the Shapes Look Like

#

The Parabola: $y = a(x - p)^2 + q$

#

• Turning point is always at $(p;\, q)$.
• Axis of symmetry is $x = p$ (the vertical line through the turning point).
• $a > 0$: opens upward (happy face). $a < 0$: opens downward (sad face).

The Hyperbola: $y = \frac{a}{x - p} + q$

#

• Vertical asymptote at $x = p$ — the graph NEVER crosses this line.
• Horizontal asymptote at $y = q$ — the graph approaches but never reaches this value.
• $a > 0$: branches in quadrants I and III (relative to the asymptote intersection). $a < 0$: quadrants II and IV.

The Exponential: $y = ab^{x-p} + q$

#

• Horizontal asymptote at $y = q$ — the curve gets close but never touches it.
• $b > 1$ and $a > 0$: growth (rises steeply to the right).
• $0 < b < 1$ and $a > 0$: decay (falls towards the asymptote).
• $a < 0$: the curve is reflected below the asymptote.

The Sketching Checklist (Works for ALL Three)

#

For every function you sketch, find and label:

1. ✅ Asymptotes (draw as dashed lines and label them)
2. ✅ Intercepts — $y$-intercept (let $x = 0$), $x$-intercept(s) (let $y = 0$)
3. ✅ Turning point (parabola only) or key point near the asymptotes
4. ✅ Shape — use $a$ to determine orientation (up/down, which quadrants)
5. ✅ Domain and Range — write them in set or interval notation
6. ✅ One extra point for accuracy

Deep Dives (click into each)

#

Each function gets its own dedicated page with full worked examples, finding equations from graphs, and exam strategies:

• The Parabola — turning point form, completing the square, 3 methods for finding equations, axis of symmetry, domain & range
• The Hyperbola — shifted asymptotes, lines of symmetry, quadrant placement, finding equations
• The Exponential — growth vs decay, finding $a$ and $b$, reflection, x-intercept existence

🚨 Common Mistakes Across All Functions

#

1. The $p$-sign trap: $(x + 2)$ means shift LEFT 2, not right. The sign flips.
2. Forgetting asymptote labels: In exams, draw asymptotes as dashed lines and write their equations. Unlabelled asymptotes = lost marks.
3. Domain/Range confusion: Domain = $x$-values (horizontal). Range = $y$-values (vertical). For the hyperbola, domain excludes $p$; for the exponential, range is restricted by $q$.
4. Drawing through asymptotes: The curve approaches but NEVER touches or crosses an asymptote.
5. Not finding enough points: The y-intercept alone isn’t enough. Calculate at least 2–3 points to draw an accurate curve.

🔗 Related Grade 11 topics:

• Quadratic Equations — x-intercepts of the parabola come from solving $ax^2 + bx + c = 0$
• Surds & Exponential Equations — exponential equations connect to graphing
• Finance, Growth & Decay — compound growth IS an exponential function

📌 Grade 10 foundation: Sketching Graphs (Linear, Quadratic, Hyperbola)

📌 Grade 12 extension: Functions & Inverses — inverses, domain restriction, and logarithms

⏮️ Number Patterns | 🏠 Back to Grade 11 | ⏭️ Finance, Growth & Decay