IB Maths AA SL 2 Functions Questions

Practise IB Maths AA SL functions through graphs, transformations, equations, inverses and models, explaining structure and selecting exact methods.

Exam points

Construct and interpret function graphs using intercepts, extrema, asymptotes, intersections, ranges and other labelled key features.
Evaluate, compose and invert functions while checking domain, range, one-to-one restrictions and the symmetry of a function and its inverse.
Solve equations and inequalities analytically or graphically, including parameter, branch and real-context restrictions.
Apply ordered translations, reflections and horizontal or vertical stretches to derive, sketch or evaluate a transformed graph.

Covered syllabus points

2.1 Functions - SL content

[Maximum number: 3]

Consider the function $f(x)=\frac{2}{\pi} \sin (3 \pi x)+2$ , where $0 \leq x \leq 2$ . The following diagram shows the graph of f.

(a)

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

The line $L_{1}$ is normal to the graph of f at point A(1,2).
The line $L_{2}$ is tangent to the graph of f at A .

[ 1 ]

$\quad\left(m_{L_{1}}=\right) 0.167\left(=\frac{1}{6}\right)$

(b)

Find the coordinates of B.

The shaded region is enclosed by the graph of f and the line $L_{1}$ between A and B .

[ 2 ]

equating $L_{1}$ and f

[Maximum number: 4]

Let $f(x)=\cos 2 x$ and $g(x)=2 x^{2}-1$ .

(a)

Find $f\left(\frac{\pi}{2}\right)$ .

[ 2 ]

$\quad f\left(\frac{\pi}{2}\right)=\cos \pi$

=-1

(b)

Find $(g \circ f)\left(\frac{\pi}{2}\right)$ .

[ 2 ]

$\quad(g \circ f)\left(\frac{\pi}{2}\right)=g(-1)\left(=2(-1)^{2}-1\right)$

[Maximum number: 4]

The velocity of a particle in $\mathrm{ms}^{-1}$ is given by $v=\mathrm{e}^{\sin t}-1$ , for $0 \leq t \leq 5$ .

(a)

On the grid below, sketch the graph of v.

Figure for Question (a) — IB Maths AA SL

[ 3 ]

(b)

Write down the positive t-intercept.

[ 1 ]

$\quad t=\pi$ (exact), 3.14

[Maximum number: 15]

Let $f(x)=3 x^{2}-6 x+p$ . The equation f(x)=0 has two equal roots.

(a)

Write down the value of the discriminant.

[ 1 ]

correct value 0 , or $36-12 p \quad \boldsymbol{A}^{2}$

(b)

Hence, show that p=3.

[ 2 ]

correct equation which clearly leads to $p=3 \quad \boldsymbol{A} \mathbf{1}$
eg $\quad 36-12 p=0,36=12 p$
p=3
AG N0

(c)

The graph of f has its vertex on the x-axis.

Find the coordinates of the vertex of the graph of f.

[ 4 ]

METHOD 1
valid approach
eg $\quad x=-\frac{b}{2 a}$
correct working A1
eg $\quad-\frac{(-6)}{2(3)}, x=\frac{6}{6}$
correct answers
eg $\quad x=1, y=0 ;(1,0)$
METHOD 2
valid approach
eg f(x)=0, factorisation, completing the square
correct working
eg $\quad x^{2}-2 x+1=0,(3 x-3)(x-1), f(x)=3(x-1)^{2}$
correct answers
eg $\quad x=1, y=0 ;(1,0)$
METHOD 3
valid approach using derivative
eg $\quad f^{\prime}(x)=0,6 x-6$
correct equation A1
eg $\quad 6 x-6=0$
correct answers
eg $\quad x=1, y=0 ;(1,0)$

(d)

Write down the solution of f(x)=0.

[ 1 ]

(e)

The function can be written in the form $f(x)=a(x-h)^{2}+k$ . Write down the value of

[ 3 ]

(i)

[ 1 ]

$\quad a=3$

A1 N1

(ii)

[ 1 ]

$\quad h=1$

A1 N1

(iii)

[ 1 ]

(f)

The graph of a function g is obtained from the graph of f by a reflection of f in the x-axis, followed by a translation by the vector $\binom{0}{6}$ . Find g, giving your answer in the form $g(x)=A x^{2}+B x+C$ .

[ 4 ]

attempt to apply vertical reflection
eg $\quad-f(x),-3(x-1)^{2}$ , sketch
attempt to apply vertical shift 6 units up
eg -f(x)+6, vertex (1,6)
transformations performed correctly (in correct order)
eg $\quad-3(x-1)^{2}+6,-3 x^{2}+6 x-3+6$ $g(x)=-3 x^{2}+6 x+3$

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)

Question 4

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)

Question (i)

Question (ii)

Question (iii)

Question (f)