SL 2.3—Graphing: IB DP Math AA Notes (HL/SL)

What Does It Mean to Sketch a Graph?

In IB Mathematics, you will often be asked to sketch a graph rather than plot it point by point. A sketch is a freehand drawing that captures the essential behaviour and key features of a function — it doesn't need to be perfectly to scale, but it must be mathematically accurate in its shape and labelling.

Sketch

A freehand graph showing the key features of a function (intercepts, turning points, asymptotes, end behaviour) without requiring precise measurements or plotting every point.

Draw

A precise, accurate graph produced using appropriate tools (e.g., ruler, set square), where coordinates and scales must be exactly correct.

Warning

In an exam, always read the instruction carefully. "Sketch" and "draw" are not interchangeable. If you are asked to sketch, you are expected to show key features clearly labelled — but if you spend time plotting dozens of points, you are wasting precious time.

The key features you should always consider when sketching are:

$x$ -intercepts (roots / zeros)
$y$ -intercept
Turning points (local maxima and minima)
Asymptotes (vertical, horizontal, or oblique)
End behaviour — what happens as $x \to + \infty$ and $x \to - \infty$
Domain and range — the set of valid inputs and resulting outputs

A Step-by-Step Approach to Sketching

Follow this structured method whenever you need to sketch a function:

Identify the function type — Is it linear, quadratic, cubic, exponential, trigonometric, rational? Knowing the family immediately tells you the general shape.
Find the $y$ -intercept — Substitute $x = 0$ .
Find the $x$ -intercept(s) — Set $f (x) = 0$ and solve.
Find turning points — Use the vertex formula $x = - \frac{b}{2 a}$ for quadratics. For other functions, you will use $f^{'} (x) = 0$ once you study differential calculus in Topic 5 — for now, use symmetry or technology.
Identify asymptotes — Particularly important for rational, exponential, and logarithmic functions.
Consider end behaviour — What does the graph do as $x \to \pm \infty$ ?
State the domain and range — Identify any restrictions on inputs or outputs.
Draw axes, plot key features, and connect with a smooth curve.

Exam Tip

You don't need calculus to sketch many common functions at SL level. For quadratics, the vertex formula $x = - \frac{b}{2 a}$ is your best friend. For exponentials and logs, remember the key intercept and asymptote.

Example

Sketching $f (x) = x^{2} - 4 x + 3$

Step 1: Quadratic — parabola opening upward (positive leading coefficient).

Step 2: $y$ -intercept: $f (0) = 0 - 0 + 3 = 3$ → point $(0, 3)$

Step 3: $x$ -intercepts: $x^{2} - 4 x + 3 = 0 \Rightarrow (x - 1) (x - 3) = 0$ → points $(1, 0)$ and $(3, 0)$

Step 4: Vertex: $x = - \frac{- 4}{2 ( 1 )} = 2$ , $f (2) = 4 - 8 + 3 = - 1$ → vertex at $(2, - 1)$

Step 5: No asymptotes for a polynomial.

Step 6: As $x \to \pm \infty$ , $f (x) \to + \infty$ .

Step 7: Domain: all real numbers $x \in R$ . Range: $f (x) \geq - 1$ , i.e., $[- 1, + \infty)$ .

Sketch: Plot $(0, 3)$ , $(1, 0)$ , $(3, 0)$ , and the minimum $(2, - 1)$ . Draw a U-shaped parabola through these points.

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What Does It Mean to Sketch a Graph?

A Step-by-Step Approach to Sketching

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