The Master Reference#
This page is your one-stop reference for every function type in Grade 12. Use it for quick revision before exams.
1. All Functions at a Glance#
| Linear | Quadratic | Hyperbola | Exponential | Logarithmic | |
|---|---|---|---|---|---|
| General Form | $y = a(x-p) + q$ | $y = a(x-p)^2 + q$ | $y = \frac{a}{x-p} + q$ | $y = ab^{x-p} + q$ | $y = a\log_b(x-p) + q$ |
| Shape | Straight line | U-shape (parabola) | Two branches | J-curve | Slow curve |
| Domain | $x \in \mathbb{R}$ | $x \in \mathbb{R}$ | $x \in \mathbb{R}, x \ne p$ | $x \in \mathbb{R}$ | $x > p$ |
| Range | $y \in \mathbb{R}$ | $y \ge q$ (if $a>0$) | $y \in \mathbb{R}, y \ne q$ | $y > q$ (if $a>0$) | $y \in \mathbb{R}$ |
| $y \le q$ (if $a<0$) | $y < q$ (if $a<0$) | ||||
| Asymptotes | None | None | $x = p$ and $y = q$ | $y = q$ | $x = p$ |
| Function Type | One-to-One | Many-to-One | One-to-One | One-to-One | One-to-One |
2. The Universal Parameters#
Every function in Grade 12 is controlled by the same set of parameters. Once you understand what $a$, $p$, and $q$ do, you can handle any function.
What “$a$” Does (Shape / Reflection / Stretch)#
| Value of $a$ | Effect on ALL functions |
|---|---|
| $a > 0$ | Standard orientation |
| $a < 0$ | Reflected (flipped) |
| $\|a\| > 1$ | Stretched vertically (narrower / steeper) |
| $0 < \|a\| < 1$ | Compressed vertically (wider / flatter) |
Special cases by function:
- Linear: $a$ = gradient (steepness and direction)
- Quadratic: $a > 0$ = happy face $(\smile)$; $a < 0$ = sad face $(\frown)$
- Hyperbola: $a > 0$ = branches in Q1 & Q3; $a < 0$ = branches in Q2 & Q4 (relative to centre)
- Exponential: $a < 0$ reflects the graph below the asymptote
What “$p$” Does (Horizontal Shift)#
| Value of $p$ | Effect on ALL functions |
|---|---|
| $p > 0$ | Graph moves RIGHT |
| $p < 0$ | Graph moves LEFT |
The sign trap: In $y = (x + 3)^2$, the shift is LEFT 3, because $x + 3 = x - (-3)$, so $p = -3$.
Additional role by function:
- Quadratic: $x = p$ is the axis of symmetry
- Hyperbola: $x = p$ is the vertical asymptote
- Logarithmic: $x = p$ is the vertical asymptote
What “$q$” Does (Vertical Shift)#
| Value of $q$ | Effect on ALL functions |
|---|---|
| $q > 0$ | Graph moves UP |
| $q < 0$ | Graph moves DOWN |
Additional role by function:
- Quadratic: $q$ is the y-value of the turning point
- Hyperbola: $y = q$ is the horizontal asymptote
- Exponential: $y = q$ is the horizontal asymptote
3. Key Points Reference#
Intercepts#
| Function | y-intercept (set $x = 0$) | x-intercept (set $y = 0$) |
|---|---|---|
| Linear $y = mx + c$ | $(0; c)$ | $(-\frac{c}{m}; 0)$ |
| Quadratic $y = a(x-p)^2 + q$ | $(0; ap^2 + q)$ | Solve $a(x-p)^2 + q = 0$ |
| Hyperbola $y = \frac{a}{x-p} + q$ | $(0; -\frac{a}{p} + q)$ | $(p - \frac{a}{q}; 0)$ |
| Exponential $y = ab^{x-p} + q$ | $(0; ab^{-p} + q)$ | Solve $ab^{x-p} = -q$ |
| Logarithmic $y = a\log_b(x-p) + q$ | Only if $p < 0$ | Solve $a\log_b(x-p) + q = 0$ |
Special Points#
| Function | Special Point |
|---|---|
| Linear | y-intercept at $(0; c)$ |
| Quadratic | Turning point at $(p; q)$ |
| Hyperbola | Centre at $(p; q)$ (NOT on the graph) |
| Exponential $y = b^x$ | Always passes through $(0; 1)$ |
| Logarithmic $y = \log_b x$ | Always passes through $(1; 0)$ |
4. Inverse Functions Reference#
| Original Function | Inverse | Notes |
|---|---|---|
| $y = mx + c$ | $y = \frac{1}{m}x - \frac{c}{m}$ | Always a function (One-to-One) |
| $y = ax^2$ | $y = \pm\sqrt{\frac{x}{a}}$ | Requires domain restriction to be a function |
| $y = \frac{a}{x}$ | $y = \frac{a}{x}$ | Is its own inverse! |
| $y = b^x$ | $y = \log_b x$ | Always a function |
| $y = \log_b x$ | $y = b^x$ | Always a function |
The Swap Rule (Universal Method)#
For ANY function:
- Write the equation with $y$ and $x$.
- Swap $x$ and $y$.
- Solve for the new $y$.
- Check if the result is a function (Vertical Line Test). If not, restrict the domain.
Graphical Check#
The graph of $f^{-1}$ is always the reflection of $f$ across the line $y = x$.
If $(a; b)$ is on $f$, then $(b; a)$ is on $f^{-1}$.
5. Transformation Summary#
Starting from the basic form and applying transformations:#
| Transformation | What changes | Example |
|---|---|---|
| Vertical shift UP by $k$ | Add $k$ to the equation | $y = x^2$ → $y = x^2 + 3$ |
| Vertical shift DOWN by $k$ | Subtract $k$ from the equation | $y = x^2$ → $y = x^2 - 3$ |
| Horizontal shift RIGHT by $k$ | Replace $x$ with $(x - k)$ | $y = x^2$ → $y = (x - 3)^2$ |
| Horizontal shift LEFT by $k$ | Replace $x$ with $(x + k)$ | $y = x^2$ → $y = (x + 3)^2$ |
| Reflection in x-axis | Multiply equation by $-1$ | $y = x^2$ → $y = -x^2$ |
| Reflection in y-axis | Replace $x$ with $-x$ | $y = 2^x$ → $y = 2^{-x}$ |
| Reflection in $y = x$ | Swap $x$ and $y$ (= Inverse) | $y = 2^x$ → $y = \log_2 x$ |
| Vertical stretch by factor $k$ | Multiply equation by $k$ | $y = x^2$ → $y = 3x^2$ |
6. Domain & Range Quick Reference#
How to determine the domain:#
- Fractions: The denominator $\ne 0$. Exclude values that make it zero.
- Square roots: The expression under the root $\ge 0$.
- Logarithms: The argument $> 0$.
- Everything else: Usually $x \in \mathbb{R}$.
How to determine the range:#
- Look at the asymptote: The range excludes the asymptote value.
- Look at the turning point: For parabolas, the range starts (or ends) at $q$.
- Check the sign of $a$: This tells you if the function goes above or below.
7. Exam Strategy: Reading a Graph#
When given an unknown graph in an exam, identify it in 3 steps:
- Shape: Straight line? U-shape? Two branches? J-curve? → Identifies the function type.
- Asymptotes: Read them off the graph → Gives you $p$ and $q$ directly.
- One point: Substitute any clear point into the equation → Solves for $a$ (or $b$).
| If you see… | It’s a… | Write the form… |
|---|---|---|
| Straight line | Linear | $y = mx + c$ |
| U-shape or ∩-shape | Quadratic | $y = a(x-p)^2 + q$ |
| Two separate branches with asymptotes | Hyperbola | $y = \frac{a}{x-p} + q$ |
| J-curve with one horizontal asymptote | Exponential | $y = ab^{x-p} + q$ |
| Slow curve with one vertical asymptote | Logarithmic | $y = a\log_b(x-p) + q$ |
💡 The Golden Rule#
Every function in Grade 12 is just the basic version ($y = x$, $y = x^2$, $y = \frac{1}{x}$, $y = b^x$, $y = \log_b x$) with three transformations applied: a stretch/reflection ($a$), a horizontal shift ($p$), and a vertical shift ($q$). If you master what $a$, $p$, and $q$ do, you master all functions.