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

[Maximum number: 5]

Line $L_{1}$ is tangent to the graph of a function f(x) at the point P(3,-1). Line $L_{2}$ is given by the equation $y=-\frac{1}{2} x-\frac{5}{2}$ and is perpendicular to $L_{1}$ .

(a)

Write down the gradient of $L_{1}$ .

[ 1 ]

(b)

Find the equation of $L_{1}$ in the form y=m x+c.

[ 2 ]

attempt to substitute their part (a) and point (3,-1) into the slope-intercept form or point-slope form of an equation
$-1=2 \times 3+c \quad$ OR $\quad y+1=2(x-3)$
y=2 x-7
Note: Equation must be in the form y=m x+c for A1 to be awarded.

(c)

Show that $L_{2}$ is not the line that is normal to f(x) at point P .

[ 2 ]

METHOD 1
attempt to show that P does not lie on $L_{2}$
e.g. $-\frac{1}{2}(3)-\frac{5}{2} \quad \mathbf{OR}$ graph showing $L_{2}$ and P in approximate correct locations
$-1 \neq-\frac{1}{2}(3)-\frac{5}{2}(-1 \neq-4) \mathbf{OR}(3,-1)$ does not lie on the graph of $L_{2}$
hence $L_{2}$ is not the normal line to f(x) at point P

METHOD 2
attempt to find the equation of the normal line at (3,-1)
$\left(-1=-\frac{1}{2}(3)+c \quad\right.$ OR $\left.\quad y+1=-\frac{1}{2}(x-3)\right)$
the normal line is $y=-\frac{1}{2} x+\frac{1}{2}$
hence $L_{2}$ is not the normal line to f(x) at point P

METHOD 3
attempt to find the intersection of $L_{1}$ and $L_{2}$

Intersection of y=2 x-7 and $y=-\frac{1}{2} x-\frac{5}{2}$ is (1.8,-3.4)
$x=1.8 \neq 3 \quad$ OR $\quad y=-3.4 \neq-1$
hence $L_{2}$ is not the normal line to f(x) at point P

Note: Accept equivalent written arguments provided values are seen.
Methods 1 and 2 are independent of the answers in (a) and (b) but FT marks can be given for Method 3.

[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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Covered syllabus points

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Question 2

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Question 4

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