Graphs

Polynomial. $y = ax^2 + bx + c$ (parabola), $y = ax^3 + \ldots$ (cubic), etc. End-behaviour controlled by leading term.

Rational. $y = \dfrac{ax + b}{cx + d}$:

• VERTICAL asymptote at $x = -\dfrac{d}{c}$ (denominator zero).
• HORIZONTAL asymptote at $y = \dfrac{a}{c}$ (ratio of leading coefficients).
• $x$-intercept where numerator = 0; $y$-intercept at $x = 0$.

Exponential. $y = a \cdot b^x$ with $b > 0$, $b \ne 1$:

• Passes through $(0, a)$.
• HORIZONTAL asymptote $y = 0$.
• $b > 1$ → growth; $0 < b < 1$ → decay.

Logarithmic. $y = \log_a x$ ($a > 1$): the INVERSE of $y = a^x$. So:

• Passes through $(1, 0)$.
• VERTICAL asymptote $x = 0$.
• Domain $x > 0$; range $\mathbb{R}$.

Starting from $y = f(x)$:

Transformation	New equation	Effect
Translation	$y = f(x - h) + k$	shift RIGHT by $h$, UP by $k$
Vertical stretch	$y = a f(x)$	factor $a$ from the $x$-axis
Horizontal stretch	$y = f(b x)$	factor $\dfrac{1}{b}$ from the $y$-axis
Reflection in $x$-axis	$y = -f(x)$	flip top-bottom
Reflection in $y$-axis	$y = f(-x)$	flip left-right

KEY caution. Inside-the-bracket changes are 'opposite' of what they look like:

• $f(x - 3)$ → shift RIGHT by 3 (not left).
• $f(2x)$ → COMPRESS horizontally by factor 2 (not stretch).

Worked example. Describe $y = -2 f(x - 3) + 5$ as a sequence of transformations of $y = f(x)$.

1. Translate right by 3 (inside-bracket).
2. Vertical stretch factor 2 (outside).
3. Reflect in the $x$-axis (negative sign).
4. Translate up by 5 (outside).

(Order: do INSIDE-bracket operations FIRST, then outside.)

Worked example. The graph of $y = f(x)$ has a turning point at $(2, -1)$. What is the corresponding point on $y = f(x + 3) - 4$?

Inside: $x \to x - 3$ relative to original $f$ — point becomes $(2 - 3, -1) = (-1, -1)$.

Outside: $y \to y - 4$ — point becomes $(-1, -5)$.

Always work the point through each transformation in order.

Intercepts.

• $y$-intercept: set $x = 0$.
• $x$-intercept(s) (zero(s)): set $y = 0$.

Asymptotes.

• VERTICAL: where the function is undefined (e.g. denominator = 0 in rationals; argument = 0 in $\ln$).
• HORIZONTAL: limit as $x \to \pm \infty$. Rationals: ratio of leading coefficients. Exponentials: $y = 0$ (or whatever constant is added).

Worked example (rational). Sketch $y = \dfrac{2x - 3}{x + 1}$.

• $y$-intercept: $x = 0 \Rightarrow y = -3$.
• $x$-intercept: $2x - 3 = 0 \Rightarrow x = \tfrac{3}{2}$.
• Vertical asymptote: $x = -1$ (where denominator = 0).
• Horizontal asymptote: $y = \dfrac{2}{1} = 2$.

Sketch with two branches: one in each region created by the asymptotes.

Intersection of two graphs. Solve $f(x) = g(x)$. Algebraically when possible (Paper 1) or with GDC 'intersect' (Paper 2).

Worked example. $f(x) = x^2$ and $g(x) = x + 6$ intersect where $x^2 = x + 6 \Rightarrow x^2 - x - 6 = 0 \Rightarrow (x - 3)(x + 2) = 0$. Points: $(3, 9)$ and $(-2, 4)$.