IB Maths AI SL 2 Functions Questions

Practise IB Maths AI SL functions through graphs, regression, transformations, equations and technology-supported models, explaining results in context.

Exam points

Use function notation, domains, ranges, composition and inverse relationships to represent inputs, outputs and model mappings.
Construct and transform graphs, then interpret intercepts, extrema, symmetry, asymptotes, intersections and other defining features.
Select, fit and use function models to calculate predictions, thresholds, optimisation quantities and parameter values within a valid domain.
Interpret model parameters such as roots, vertices, amplitudes, periods, asymptotes, phase shifts, carrying capacity and half-life.
Evaluate assumptions, domain choices, extrapolation risk and contextual reasonableness using data, graph behaviour and units.

Covered syllabus points

2.1 Functions - SL content

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Study 2 Functions conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

[Maximum number: 10]

The mean annual temperatures for Earth, recorded at fifty-year intervals, are shown in the table.

Tami creates a linear model for this data by finding the equation of the straight line passing through the points with coordinates (1708,8.73) and (1958,9.45).

(a)

Calculate the gradient of the straight line that passes through these two points.

[ 2 ]

$\frac{9.45-8.73}{1958-1708}$

=0.00288\left(\frac{9}{3125}\right)

(b)

Interpret the meaning of the gradient in the context of the question.

[ 1 ]

(c)

State appropriate units for the gradient.

[ 1 ]

$\quad{ }^{\circ} \mathrm{C} /$ year OR degrees C per year A1

Note: Do not follow through from part (b)(i) into (b)(ii).

(d)

Find the equation of this line giving your answer in the form y=m x+c.

[ 2 ]

attempt to substitute point and gradient into appropriate formula $8.73=0.00288 \times 1708+c \Rightarrow c=3.81096 \ldots$
or $9.45=0.00288 \times 1958+c \Rightarrow c=3.81096$ .
equation is y=0.00288 x+3.81

(e)

Use Tami's model to estimate the mean annual temperature in the year 2000.

[ 2 ]

attempt to substitute 2000 into their part (c)
$0.00288 \times 2000+3.81096 \ldots$ $=9.57\left({ }^{\circ} \mathrm{C}\right)(9.57096 \ldots)$

(f)

Thandizo uses his regression line to predict the year when the mean annual temperature will first exceed $15^{\circ} \mathrm{C}$ .

State two reasons why Thandizo's prediction may not be valid.

[ 2 ]

[Maximum number: 6]

A function is defined by $f(x)=\frac{4}{x}-3$ , for $-6 \leq x \leq 10, x \neq 0$ .

(a)

Find f(10).

[ 1 ]

$-2.6\left(\frac{-13}{5}\right)$

(b)

Find the range of f.

[ 3 ]

EITHER
$(-\infty,-3.67] \cup[-2.6, \infty)$
OR
$y \leq-3.67$ or $y \geq-2.6$
Note: Award A1 for $-3.67\left(-3.66666 \ldots,-\frac{11}{3}\right)$ seen
A1 for their $-2.6\left(-\frac{13}{5}\right)$ seen and both sets of inequalities/brackets correct;
A1 if completely correct, including union symbol or "or".
It is possible to award A0A1A0 for correctly presenting their -3.67 (and their -2.6 ).
The -3.67 maybe seen as part of a sketch.

Award A1A0A0 for $-3.67 \leq y \leq-2.6$ or [-3.67,-2.6].

(c)

Find $f^{-1}(-2)$ .

[ 2 ]

setting f(x) equal to -2 f(x)=-2 OR $-2=\frac{4}{x}-3$ OR sketch of graph with line labelled -2

[Maximum number: 2]

A hollow chocolate box is manufactured in the form of a right prism with a regular hexagonal base. The height of the prism is $h \mathrm{~cm}$ , and the top and base of the prism have sides of length $x \mathrm{~cm}$ .

Sketch the graph of $V=300 \sqrt{3} x-\frac{9}{4} x^{3}$ , for $0 \leq x \leq 16$ .

[Maximum number: 8]

The Scheveningen Ferris wheel's lowest point is 8 m above sea level, and its highest point is 45 m above sea level.

(a)

Find the value of b.

[ 1 ]

$\quad\left(b=\frac{360}{15}=\right) 24$

(b)

Find the value of d.

[ 2 ]

$\frac{\max +\min }{2}$

\begin{aligned} & (d=) \frac{45+8}{2} \text { OR } 45-18.5 \text { OR } 8+18.5 \\ & =26.5 \end{aligned}

(c)

Hence, write down the equation of the sinusoidal model.

[ 2 ]

$h(t)=18.5 \sin (24 t)+26.5$

Note: Award A1 for 18.5 seen as parameter a, A1 for a completely correct equation (including LHS).

(d)

Use the model to find the values of t when the height of this pod is 33 m above sea level for $0 \leq t \leq 15$ .

Since the Ferris wheel opened, it has been operating for 3000 days, and each day it rotates nonstop for 8 hours.

[ 3 ]

attempt to equate 33 with h(t) OR sketch graph of curve and line
$h(t)(=18.5 \sin (24 t)+26.5)=33$ $t=0.857(0.857083 \ldots)$ and $t=6.64(6.64291 \ldots)$ (minutes)

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