7. Curve Sketching & Related Graphs

Video coming soon2 worked examples

Theory

A full curve sketch combines intercepts, stationary points, asymptotes and behaviour at infinity. Related graphs are obtained from $y = f(x)$ by transformations: $f(x) + a$ shifts up, $f(x + a)$ shifts left, $af(x)$ stretches vertically, $-f(x)$ reflects in the $x$ -axis, $|f(x)|$ reflects the negative parts upward, and $y = f^{-1}(x)$ reflects in the line $y = x$ .

The Golden Rule: apply one transformation at a time and track how the key features — roots, turning points and asymptotes — move.

⚠️ Common Examiner Traps

Horizontal shift direction: $f(x + a)$ moves the graph left for $a > 0$ — the opposite of what many expect.
Modulus graph: $|f(x)|$ reflects only the parts below the $x$ -axis; the rest is unchanged.
Derivative graph: the roots of $y = f'(x)$ are the $x$ -values of the turning points of $y = f(x)$ .

Worked examples

Example 1

The graph of $y = f(x)$ has a maximum turning point at $(2, 5)$ . State the coordinates of the corresponding turning point on the graph of $y = f(x - 1) + 3$ .

Step 1: The $f(x-1)$ part shifts the graph $1$ unit to the right; the $+3$ shifts it $3$ units up.

Step 2: Apply both shifts to the point $(2, 5)$ :

(2 + 1,\ 5 + 3) = (3, 8)

The turning point is a maximum at $(3, 8)$ .

Example 2

The graph of $y = f(x)$ has stationary points at $x = 2$ and $x = 4$ . State the $x$ -coordinates where the graph of $y = f'(x)$ crosses the $x$ -axis.

Step 1: A stationary point of $f$ is where $f'(x) = 0$ , i.e. where the derivative graph meets the $x$ -axis.

Step 2: So $y = f'(x)$ crosses the $x$ -axis at $x = 2$ and $x = 4$ .