IB Mathematics AA HL 2.1 Functions Question Bank

Practise IB Mathematics AA HL 2.1 by solving advanced function, transformation, inverse, composition and modelling problems with exact reasoning.

Exam points

HL syllabus concepts and methods
HL extended reasoning

Covered syllabus points

SL 2.1—Straight lines SL 2.2—Functions and inverses SL 2.3—Graphing functions SL 2.4—Key graph features SL 2.5—Composite and inverse functions SL 2.6—Quadratic functions SL 2.7—Quadratic equations and inequalities SL 2.8—Reciprocal and rational functions SL 2.9—Exponential and logarithmic functions SL 2.10—Solving equations SL 2.11—Graph transformations

[Maximum number: 4]

This question investigates a ratio of lengths found from the line passing through the points of inflexion of a quartic curve of the form $y=x^{4}-m x^{3}+n x$ .
The curve $y=x^{4}-3 x^{3}+3 x$ has points of inflexion at B and C . The line passing through B and C intersects the curve again at points A and D . This is shown in the following graph.

Figure for Question 2.1 Functions - SL content question 1 — IB Maths AA HL

(a)

Show that the equation of the line through B and C is y=-0.375 x.

[ 2 ]

$\quad y$ -intercept =0 (as line passes through (0,0) )

Note: Award R1 for correctly substituting point B or point C to show that y=0.

Award R1 for y=-0.375 x+0.

Note: Award AOFT if their answer to (b) doesn't lead to the given answer, but condone $\frac{-0.563}{1.5}$ .

so equation is y=-0.375 x
AG
Note: Award at most A1 R0 for working backwards to verify points B and C lie on the given line.

(b)

Show that the equation of the line through B and C is $y=\left(-\frac{m^{3}}{8}+n\right) x$ .

[ 2 ]

attempt to divide their $y_{C}$ by their $x_{C}$ OR substitute into $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

\begin{aligned} & \text { e.g. }\left(-\frac{m^{4}}{16}+\frac{n m}{2}\right) \div \frac{m}{2} \\ & \left(-\frac{m^{4}}{16}+\frac{n m}{2}\right) \times \frac{2}{m} \text { OR equivalent manipulation leading to given answer } \end{aligned}

Note: Award AO FT if their answer does not lead to the AG.

=-\frac{m^{3}}{8}+n \text { and } y \text {-intercept }=0

\text { so equation is } y=\left(-\frac{m^{3}}{8}+n\right) x

Note: Award at most M1A0 for working backwards to verify points B and C lie on the given line.

[Maximum number: 1]

A function f is defined by $f(x)=\arcsin \left(\frac{x^{2}-1}{x^{2}+1}\right), x \in \mathbb{R}$ .

State the domain of $g^{-1}$ .

domain is $-\frac{\pi}{2} \leq x<\frac{\pi}{2}$
Note: Accept correct alternative notations, for example, $\left\lfloor-\frac{\pi}{2}, \frac{\pi}{2}\right\rfloor$ or $\left\lfloor-\frac{\pi}{2}, \frac{\pi}{2}\right)$ .

Marking guidance:

Accept [-1.57,1.57[ if correct to 3 s.f.

[Maximum number: 2]

The following question explores features of a family of curves. The family is then linked to a homogeneous differential equation.
Consider the curve given by $y=\frac{x\left(x^{2}-16\right)}{x^{2}+16}$ .

(a)

Sketch the curve of y for $-10 \leq x \leq 10$ .

[ 1 ]

(b)

State the coordinates of the points where the curve crosses the x-axis.

[ 1 ]

$\quad(-4,0),(0,0),(4,0)$

Note: Award A0 if additional points are given.

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Covered syllabus points

2.1 Functions - SL content question 1

Question (a)

Question (b)

2.1 Functions - SL content question 2

2.1 Functions - SL content question 3

Question (a)

Question (b)