IB Maths AA SL 2 Functions Question Bank

Practise interpreting functions through equations, graphs, intercepts, inverses, asymptotes and model values in algebraic contexts.

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.

Figure for Question 2 Functions question 1 — IB Maths AA SL

(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
A1A1
N2
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
A1A1
N2
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
A1A1
N2
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

2 Functions question 1

Question (a)

Question (b)

2 Functions question 2

Question (a)

Question (b)

2 Functions question 3

Question (a)

Question (b)

2 Functions question 4

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)

Question (i)

Question (ii)

Question (iii)

Question (f)