IB Maths AA HL 2.2 Functions Ahl Content Questions

Practise IB Mathematics AA HL 2.2 by solving advanced function models, complex transformations, differential behaviour and parameter-sensitive equations.

Exam points

Use factor/remainder theorems and sums/products of roots to determine polynomial coefficients, factors, zeros or complex-root conditions.
Analyse further rational functions by deriving vertical, horizontal or oblique asymptotes, intercepts, extrema, ranges and parameter restrictions.
Test odd/even or self-inverse properties algebraically, and construct inverses with the domain restriction needed for one-to-one behaviour.
Solve inequalities between functions by finding boundary intersections and selecting valid intervals with polynomial or technology methods.
Apply absolute-value, reciprocal, square and inner-argument transformations to graphs, then solve the resulting modulus equations or inequalities.

Covered syllabus points

AHL 2.12 (HL)—Polynomial functions AHL 2.13 (HL)—Further rational functions AHL 2.14 (HL)—Odd/even and inverse functions AHL 2.15 (HL)—Inequalities between functions AHL 2.16 (HL)—Further graph transformations

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Study 2.2 Functions - AHL content conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

[Maximum number: 1]

Consider a three-digit code a b c, where each of a, b and c is assigned one of the values 1,2,3,4 or 5 .

Express P(x) as a product of linear factors.

$x^{3}+4 x^{2}+5 x+2=\left(x^{2}+3 x+2\right)(x+1)$

=(x+2)(x+1)(x+1)

[Maximum number: 3]

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)

State whether the function $f(x)=\frac{x\left(x^{2}-16\right)}{x^{2}+16}$ is odd, even or neither. Justify your answer. [2]

Now consider the general curve given by $y=\frac{x\left(x^{2}-A\right)}{x^{2}+A}$ , where A is a positive constant
and $x \in \mathbb{R}$ . and $x \in \mathbb{R}$ .

[ 2 ]

EITHER
odd A1
since $f(-x)=\frac{(-x)\left((-x)^{2}-16\right)}{(-x)^{2}+16}=\frac{-x\left(x^{2}-16\right)}{x^{2}+16}=-f(x)$ (or equivalent) $\quad \boldsymbol{R} 1$
OR
odd A1
since graph of f has rotational symmetry order 2 (of 180 degrees) about $(0,0) \quad \boldsymbol{R}$ 1

OR
odd A1 since graph has reflection symmetry over the x-axis followed by reflection symmetry over the y-axis (or vice versa)

R1

Note: Do not award A1R0.

(b)

Hence, determine the equation of the oblique asymptote to the curve.

[ 1 ]

hence oblique asymptote is y=x (since $\frac{2 A x}{x^{2}+A} \rightarrow 0$ as $x \rightarrow \pm \infty$ )

Note: Award AO if not given as an equation.

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Question 2

Question (a)

Question (b)