Sketch on a domain with the GDC

Sketch a function on a given domain with the GDC

In 6 of 20 recent exams
Paper 2, calculator
2 to 4 marks

How it is set in the exam

Paper 2 (calculator) only: a 3-mark sketch, often after 2–4 marks of preparation parts (the values at the ends, then the zeros), usually in Section A (5 of the 6 versions so far; once as the first part of a Section B question). Common (6 of 20 exam versions). The same sketch skills earn marks in every other sketching type (the table in Sketching).

Spot it in the exam

Exam phrases to spot (4)

“sketch the graph of $y = f(x)$ for $-2 \le x \le 3$ ”
“on the axes provided, sketch the graph of $f$ , showing the values of any axis intercepts and the coordinates of any local maximum or minimum points”
“find $f(0)$ and $f(8)$ … find the zeros of $f$ … sketch the graph of $f$ ”
“label the endpoints”

In disguise (2)

the function is built in an earlier part: form $f(g(x))$ , then sketch it (Form a composite function and use it works that chain)
the sketch is only a tool for a later part, such as "hence write down the range" or "hence find the number of solutions" (Solve graphically)

Variants (6)

the features: the zeros, a maximum and the ends; or the zeros, the $y$ -intercept, a minimum and the ends
a function symmetric about the $y$ -axis: zeros in a $\pm$ pair and a turning point on the axis
a root or a log minus a linear term: one hump, usually with two zeros
a polynomial of a trig function, in radians
an exponential times a sine or cosine whose zeros are the two ends of the domain: one hump that leans toward the side where the exponential is larger; it can open a Section B question that asks for the derivative only later
the end values and the zeros asked for first, as separate parts

Looks like it, but isn't (3)

the other "sketch" questions are in the table in Sketching. The closest look-alikes:

the turning points are found with $f'$ and their nature is justified (often on ): Stationary points
$\frac{ax + b}{cx + d}$ by hand: Rational functions; a parabola: Find the vertex
$f^{-1}$ from the graph of $f$ : The graphs of f and f⁻¹; a velocity: Velocity and acceleration

Worked example

The ends and the zeros first, then the sketch

Paper 2 (calculator)

Question

Let $f(x) = 8\ln(x + 1) - x - 3$ , for $0 \le x \le 30$ .
(a) Find (i) $f(0)$ (ii) $f(30)$ . [2]
(b) Find the zeros of $f$ . [2]
(c) Sketch the graph of $f$ , showing the endpoints, the zeros and the coordinates of the maximum point. [3]

Try it yourself first, then compare.

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Write your working on paper or on screen. Paper Zero marks it step by step, like the worked example, and shows where each mark went. Free, with no account.

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Steps

Graph $f$ on the given domain
(graph a function on a given domain), in radians for any trig. Look at the whole picture first: how many humps and dips, how many zeros, where the ends are.
Find and write down each key feature,
to 3 s.f., or exactly when the value is exact (such as $f(0) = -3$ in Example 2.1, or the zero $x = 0$ in Example 2.2): the end values $f(a)$ and $f(b)$ (define a function and evaluate it), the zeros (zeros and intersections), the $y$ -intercept, and each turning point (maximum or minimum). With symmetry, find one half and mirror it.
Draw the axes
with scales that fit every feature. Plot and label each feature, then join them with the shape from the screen.
Stop at the ends,
with a closed dot for an included end. Check the heights against each other: is an end higher or lower than a turning point?

Common mistakes

Features with no shape
labelled points joined by the wrong shape (or not joined) lose the shape mark where there is one, and often the zero and turning-point marks that depend on the shape.
A curve that runs past the ends,
or ends left unlabelled: the endpoint mark is lost.
Coordinates written beside the graph but not marked on it
they score only where the point is placed.

More mistakes (4)

A zero missed where the graph touches the axis
(the origin in Example 2.2): count the zeros on the graph, not just the ones the zero-finder reports.
A lopsided hump drawn symmetric
an exponential times a sine or cosine leans toward the side where the exponential is larger, and the lean can have a mark of its own (Example 2.3). Check where the top is against the middle of the domain.
Degree mode
for a trig function, or $\log$ typed for $\ln$ : a completely different graph.
A local maximum taken as the greatest value
an end can be higher. It matters as soon as a later part asks for the range.

Easy to confuse with

Next typeSolve graphically

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Easy to confuse with

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