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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.

Syllabus
First assessment 2021
Course
Mathematics: analysis and approaches HL
Level
HL

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 functionsAHL 2.13 (HL)—Further rational functionsAHL 2.14 (HL)—Odd/even and inverse functionsAHL 2.15 (HL)—Inequalities between functionsAHL 2.16 (HL)—Further graph transformations

Question 1

[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.

Question 2

[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=x(x2−16)x2+16y=\frac{x\left(x^{2}-16\right)}{x^{2}+16}.

Question (a)

(a)

State whether the function f(x)=x(x2−16)x2+16f(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=x(x2−A)x2+Ay=\frac{x\left(x^{2}-A\right)}{x^{2}+A}, where A is a positive constant
and x∈Rx \in \mathbb{R}. and x∈Rx \in \mathbb{R}.

[ 2 ]

Question (b)

(b)

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

[ 1 ]

Question 3

[Maximum number: 3]

Let f(x)=x4+0.2x3−5.8x2−x+4,x∈Rf(x)=x^{4}+0.2 x^{3}-5.8 x^{2}-x+4, x \in \mathbb{R}.

Find the solutions of f(x)>0.

Question 4

[Maximum number: 1]

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=x(x2−16)x2+16y=\frac{x\left(x^{2}-16\right)}{x^{2}+16}.

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

Question 5

[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=x(x2−16)x2+16y=\frac{x\left(x^{2}-16\right)}{x^{2}+16}.

State whether the function f(x)=x(x2−16)x2+16f(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=x(x2−A)x2+Ay=\frac{x\left(x^{2}-A\right)}{x^{2}+A}, where A is a positive constant
and x∈Rx \in \mathbb{R}. and x∈Rx \in \mathbb{R}.

Question 6

[Maximum number: 3]

Consider the function f(x)=ax3+bx2+cx+df(x)=a x^{3}+b x^{2}+c x+d, where x∈Rx \in \mathbb{R} and a,b,c,d∈Ra, b, c, d \in \mathbb{R}.

Consider the function g(x)=12x3−3x2+6x−8g(x)=\frac{1}{2} x^{3}-3 x^{2}+6 x-8, where x∈Rx \in \mathbb{R}.

g(x)\quad g(x) can be written in the form p(x−2)3+qp(x-2)^{3}+q, where p,q∈Rp, q \in \mathbb{R}. Find the value of p and the value of q.

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