2.3 The graph of a function (IB Maths: Analysis and Approaches HL): Revision notes

Section 1

The graph of a function

The graph of a function $f$ is the set of all points $(x, y)$ with $y = f(x)$ . Its equation is $y = f(x)$ .

A point $(a, b)$ is on the graph exactly when $f(a) = b$ . To test a point, substitute its $x$ -coordinate and compare with its $y$ -coordinate.
If a graph passes through a known point, substituting it gives an equation, which is how you find an unknown constant: $f(x) = x^2 - 4x + k$ through $(1, 2)$ gives $1 - 4 + k = 2$ , so $k = 5$ .
Where the graph meets a horizontal line $y = c$ , solve $f(x) = c$ .

Key termsgraph of a functionpoint on a graph

Common mistake

Sign slips when substituting negatives: for $x = -1$ , $-4x = +4$ , so $f(-1) = 1 + 4 + 5 = 10$ .

Exam tip

Put negative inputs in brackets on your calculator: $(-1)^2$ , not $-1^2$ .

Section 2

Describing a graph from information or a context

On paper you may be asked to sketch a graph; on this platform you describe it in words instead. A full description gives the key features:

the domain and the end points of the graph (with their coordinates),
the intercepts with the axes,
any maximum or minimum points,
any asymptotes, and whether the graph is increasing or decreasing.

In a context, label what each axis means and use units. Tank 1 with $V(t) = 800 - 50t$ , $0 \le t \le 16$ : a straight line segment from $(0, 800)$ to $(16, 0)$ , decreasing at 50 litres per minute. The domain is set by the context: the tank cannot hold negative water, so the graph stops at $t = 16$ .

Key termskey featuresend point

Exam tip

On a restricted domain the greatest or least value may be at an end point, not at a turning point.

Section 3

Using technology to graph functions

A graphic display calculator (GDC) graphs a function and finds its features for you: zeros, maximum and minimum points, and intersections. Good habits:

Choose a viewing window that fits the domain, e.g. $0 \le x \le 100$ for 100 lamps. A poor window can hide a turning point or a zero.
Write down answers to 3 significant figures unless told otherwise, and say what you did (e.g. 'zeros from GDC').
Interpret the output: $x = 9.22$ lamps is impossible, so decide whether to round up or down by checking values either side ( $P(9) < 0$ , $P(10) > 0$ ).

Key termsGDCviewing window

Common mistake

Rounding a context answer to the nearest whole number without checking: $86.8$ lamps means at most 86 for a profit, because $P(87) < 0$ .

Section 4

Sums and differences of functions

You can form new functions by adding or subtracting: $(f + g)(x) = f(x) + g(x)$ and $(f - g)(x) = f(x) - g(x)$ . On a GDC, enter $y_1 = f(x)$ , $y_2 = g(x)$ and $y_3 = y_1 - y_2$ .

Differences are especially useful in context:

Profit = revenue $-$ cost: $P(x) = R(x) - C(x)$ . Use brackets: $60x - 0.5x^2 - (400 + 12x)$ .
The gap between two quantities: $d(t) = a(t) - b(t)$ is positive when $a$ is larger and negative when $b$ is larger.
$f(x) = g(x)$ exactly where $(f - g)(x) = 0$ , so the zeros of the difference give the intersections of the two graphs.

Key termssum of functionsdifference of functions

Common mistake

Forgetting the brackets: $R(x) - 400 + 12x$ adds the variable cost instead of subtracting it.

Must know

$(a, b)$ is on $y = f(x)$ exactly when $f(a) = b$ ; use this to find unknown constants.
To find where $y = f(x)$ meets $y = c$ , solve $f(x) = c$ .
Describe a graph by its domain, end points, intercepts, maximum and minimum points and asymptotes.
On a GDC, set a sensible window and give answers to 3 s.f.
$(f - g)(x) = 0$ where the graphs of $f$ and $g$ intersect; the sign of $f - g$ tells you which is larger.
Interpret GDC answers in context (whole numbers, units, domain).

That's the notes covered.

Carry on to the next subtopic.