IB Maths AA SL 2.1 Functions Sl Content Questions

Practise IB Mathematics AA SL 2.1 by analysing function notation, domains, transformations, inverses, composite functions and graph behaviour.

Exam points

Construct and interpret function graphs by labelling intercepts, extrema, asymptotes, intersections and other stated key features.
Evaluate functions, compositions and inverses, checking domain, range, one-to-one restrictions and the symmetry of f and fâ»Â¹.
Solve quadratic, rational, exponential or logarithmic equations and inequalities, using algebraic or graphical methods and rejecting invalid branches.
Apply translations, reflections and horizontal or vertical stretches in the correct order, then derive or sketch the transformed graph.

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: 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: 1]

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.

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 .

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

[Maximum number: 2]

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

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

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

=-1

[Maximum number: 3]

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

On the grid below, sketch the graph of v.

[Maximum number: 1]

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

Write down the positive t-intercept.

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

[Maximum number: 2]

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

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

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

[Maximum number: 7]

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

(a)

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 $\quad x=1, y=0 ;(1,0)$

(b)

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 ]

(c)

Write down the value of the discriminant.

[ 1 ]

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

(d)

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

(e)

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

[ 1 ]

[Maximum number: 3]

Let $f(x)=\frac{2 x-6}{1-x}$ , for $x \neq 1$ .

(a)

For the graph of f

[ 3 ]

(i)

write down the equation of the vertical asymptote;

[ 1 ]

$\quad x=1$ (must be equation)

(ii)

find the equation of the horizontal asymptote.

[ 2 ]

valid approach
eg sketch, $\frac{2 x}{-1 x}$ , inputting large values of x, L'Hopital's rule
y=-2 (must be equation)

[Maximum number: 6]

Let $f(x)=A \mathrm{e}^{k x}+3$ . Part of the graph of f is shown below.

The y-intercept is at (0,13).

(a)

Show that A=10.

[ 2 ]

substituting (0,13) into function
e.g. $13=A e^{0}+3$

13=A+3

(b)

Given that f(15)=3.49 (correct to 3 significant figures), find the value of k.

[ 3 ]

substituting into f(15)=3.49
e.g. $3.49=10 \mathrm{e}^{15 k}+3,0.049=\mathrm{e}^{15 k}$
evidence of solving equation
e.g. sketch, using ln
$k=-0.201\left(\right.$ accept $\left.\frac{\ln 0.049}{15}\right)$

(c)

Write down the equation of the horizontal asymptote of the graph f.

Let $g(x)=-x^{2}+12 x-24$ .

[ 1 ]

$\quad y=3$

[Maximum number: 2]

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.

Find the coordinates of B.

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

equating $L_{1}$ and f

[Maximum number: 4]

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

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 $g(x)=A x^{2}+B x+C$ .

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 5

Question 6

Question 7

Question 8

Question 9

Question (a)

Question (b)

Question (i)

Question (ii)

Question (iii)

Question (c)

Question (d)

Question (e)

Question 10

Question (a)

Question (i)

Question (ii)

Question 11

Question (a)

Question (b)

Question (c)

Question 12

Question 13