Transformations - IB Math AA Question Bank

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Transformations

Practice exam-style IB Math AA questions for Transformations, aligned with the syllabus and grouped by topic.

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Question 1

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

Consider the quadratic function $h(x)=3x^2+12x+7$ , for $x\in\mathbb{R}$ .

Write $h(x)$ in the form $a(x-p)^2+q$ .

[2]

Describe the transformations that transform the graph of $y=x^2$ to the graph of $y=h(x)$ .

[3]

State the range of $h$ .

[1]

Question 2

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

Let

g(x)=\frac{5}{x+2}-1,\quad x\neq -2

The graph of $y=g(x)$ is a transformation of the graph of $y=\frac{1}{x}$ .

Give a full geometric description of the transformations from $y=\frac{1}{x}$ to $y=g(x)$ .

[3]

Write down the equations of the two asymptotes of the graph of $y=g(x)$ .

[1]

Find the value of $x$ for which $g(x)=2.5$ .

[1]

Question 3

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

Let $f(x)=\sqrt{x}$ , $x\geq 0$ . A function $g$ is defined by

g(x)=2\sqrt{x+3}-1

State the domain and range of $g$ .

[2]

Give a full geometric description of the transformations from $y=f(x)$ to $y=g(x)$ .

[2]

Solve $g(x)=5.5$ .

[1]

Question 4

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The point $P(2,-3)$ lies on the graph of $y=f(x)$ . The domain of $f$ is $[-1,5]$ and the range of $f$ is $[-4,6]$ .

A new function $g$ is defined by $g(x)=1-2f(x-3)$ .

Find the coordinates of the image of $P$ on the graph of $y=g(x)$ .

[2]

Find the domain and range of $g$ .

[2]

Describe fully the transformations that transform the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

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Question 5

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=\dfrac{1}{x}$ , $x\ne 0$ , and let $g(x)=2-\dfrac{3}{x+4}$ , $x\ne -4$ .

Describe the transformations that transform the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

State the equations of the asymptotes of the graph of $y=g(x)$ .

[2]

Solve $g(x)=5$ .

[2]

Question 6

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=e^x$ , for $x\in\mathbb{R}$ . A function $g$ is defined by $g(x)=2e^{x-1}-3$ .

Describe the transformations that transform the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

State the equation of the horizontal asymptote of the graph of $y=g(x)$ .

[1]

Solve $g(x)=1$ .

[2]

Question 7

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The graph of $y=f(x)$ has a vertical asymptote $x=1$ , a horizontal asymptote $y=-2$ , and passes through the point $A(-1,3)$ .

A new function $g$ is defined by

g(x)=2f(x-4)+5

Give a full geometric description of the transformations which map the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

Find the coordinates of the image of $A$ on the graph of $y=g(x)$ .

[1]

Find the equations of the asymptotes of the graph of $y=g(x)$ .

[2]

Question 8

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The graph of $y=p(x)$ is obtained from the graph of $y=x^2$ by a vertical stretch followed by a translation. The vertex of $y=p(x)$ is $(2,-3)$ and the graph passes through the point $(5,9)$ .

Write $p(x)$ in the form $p(x)=a(x-h)^2+k$ .

[2]

Give a full geometric description of the transformations from $y=x^2$ to $y=p(x)$ .

[2]

Find the $x$ -intercepts of the graph of $y=p(x)$ .

[1]

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Question 9

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Let $f(x)=2^x$ , for $x\in\mathbb{R}$ . A function $h$ is defined by

h(x)=3f(x-1)-7

Describe the transformations which map the graph of $y=f(x)$ to the graph of $y=h(x)$ .

[3]

Find the $y$ -intercept of the graph of $y=h(x)$ .

[1]

Solve $h(x)=0$ .

[1]

Question 10

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

f(x)=e^x+x

for $x\in\mathbb{R}$ . A function $g$ is defined by

g(x)=2f(x-1)+3

Give a full geometric description of the transformations from $y=f(x)$ to $y=g(x)$ .

[2]

Find the coordinates of the point on the graph of $y=g(x)$ with $y$ -coordinate $10$ .

[2]

The inverse function $g^{-1}$ exists. Hence find $g^{-1}(10)$ and state the corresponding point on the graph of $y=g^{-1}(x)$ .

[2]

Question 11

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The graph of

g(x)=\frac{A}{x-h}+k

is obtained from the graph of $y=\frac{1}{x}$ by transformations. It has vertical asymptote $x=3$ , horizontal asymptote $y=-2$ , and passes through the point $(5,1.2)$ .

Find the values of $h$ and $k$ .

[2]

Find the value of $A$ .

[2]

Find the coordinates of the points where the graph of $y=g(x)$ intersects the line $y=x$ .

[2]

Question 12

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=\cos x$ and $g(x)=4-2\cos(3x)$ , for $x\in\mathbb{R}$ .

Describe the transformations that transform the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

Find the period and range of $g$ .

[3]

Find the smallest positive value of $x$ for which $g(x)$ is a maximum.

[1]

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Question 13

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The graph of $y=f(x)$ has $x$ -intercepts at $(-2,0)$ and $(4,0)$ . The range of $f$ is $[-3,\infty)$ .

A function $h$ is defined by

h(x)=f\left(\frac{x-6}{2}\right)+1

Find the coordinates of the images of the two $x$ -intercepts on the graph of $y=h(x)$ .

[2]

Describe the transformations that transform the graph of $y=f(x)$ to the graph of $y=h(x)$ .

[3]

State the range of $h$ .

[1]

Question 14

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The graph of $y=f(x)$ has stationary points at $A(2,5)$ and $B(6,-1)$ . A transformed function is defined by

g(x)=a f(b(x-c))+d

where $a>0$ and $b>0$ . The images of $A$ and $B$ on the graph of $y=g(x)$ are $A'(1,8)$ and $B'(3,-4)$ respectively.

Find $b$ and $c$ .

[3]

Find $a$ and $d$ .

[2]

Hence write $g(x)$ in terms of $f$ .

[1]

Question 15

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=\arctan x$ , for $x\in\mathbb{R}$ . A function $g$ is defined by

g(x)=2\arctan(4-2x)+\frac{\pi}{3}

Describe a sequence of transformations that transforms the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[4]

State the equations of the horizontal asymptotes of the graph of $y=g(x)$ .

[2]

Find the $x$ -coordinate of the point where the graph of $y=g(x)$ crosses the line $y=\dfrac{\pi}{3}$ .

[1]

Question 16

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=\ln x$ , $x>0$ . A function $g$ is defined by

g(x)=\ln\left(\frac{x-2}{3}\right)+1

Describe the transformations that transform the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

State the domain of $g$ and the equation of its vertical asymptote.

[2]

Solve $g(x)=0$ .

[2]

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Question 17

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A function $f$ has domain $[-3,9]$ and range $[-5,4]$ . The point $(6,-3)$ lies on the graph of $y=f(x)$ .

A function $h$ is defined by

h(x)=-2f\left(\frac{x+4}{3}\right)+1

Describe the transformations which map the graph of $y=f(x)$ to the graph of $y=h(x)$ .

[2]

Find the coordinates of the image of the point $(6,-3)$ on the graph of $y=h(x)$ .

[2]

Find the domain and range of $h$ .

[2]

Question 18

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

f(x)=x^4-4x^2+1

for $x\in\mathbb{R}$ . A function $g$ is defined by

g(x)=-f(2(x-1))+3

Give a full geometric description of the transformations from the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

Find the coordinates of the local maximum points of the graph of $y=g(x)$ .

[2]

Solve $g(x)=0$ for $-1\leq x\leq 3$ .

[2]

Question 19

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

f(x)=x^3-4x

for $-3\leq x\leq 3$ . A function $g$ is defined by

g(x)=|f(x)-1|

Describe how the graph of $y=g(x)$ is obtained from the graph of $y=f(x)$ .

[2]

Solve $g(x)=2.5$ for $-3\leq x\leq 3$ .

[3]

Find the minimum value of $g(x)$ and the values of $x$ at which it occurs.

[2]

Question 20

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

f(x)=x+\ln x

for $x>0$ . For $k>0$ , a function $g$ is defined by

g(x)=f(kx)-2

The graph of $y=g(x)$ passes through the point $(3,4)$ .

Find the value of $k$ .

[2]

Describe the transformations from $y=f(x)$ to $y=g(x)$ .

[2]

Find the $x$ -intercept of the graph of $y=g(x)$ .

[2]

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Question 21

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A function $f$ has domain $[-2,4]$ and range $[-1,5]$ . The points $A(-1,2)$ and $B(3,-1)$ lie on the graph of $y=f(x)$ . A new function $g$ is defined by $g(x)=4-2f\left(\frac{x-1}{3}\right)$ .

I.

Describe fully the transformations that transform the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

II.

Find the coordinates of the images of $A$ and $B$ on the graph of $y=g(x)$ .

[2]

Find the domain and range of $g$ .

[2]

It is known that $f(x)=0$ only when $x=0$ and $x=4$ . Find the solutions of $g(x)=4$ .

[1]

Question 22

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The graph of a quadratic function $p$ is obtained from the graph of $y=x^2$ by transformations. The vertex of the graph of $y=p(x)$ is $(3,4)$ and the graph passes through $(1,0)$ .

I.

Write $p(x)$ in the form $a(x-h)^2+k$ .

[3]

II.

Describe fully the transformations that transform the graph of $y=x^2$ to the graph of $y=p(x)$ .

[3]

Solve $p(x)\geq 3$ .

[2]

A point on $y=x^2$ has coordinates $(t,t^2)$ . Find the coordinates of its image on $y=p(x)$ .

[1]

Question 23

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The function $g$ has the form $g(x)=k+\frac{a}{x-h}$ , where $a$ , $h$ and $k$ are constants. The graph of $y=g(x)$ has vertical asymptote $x=2$ , horizontal asymptote $y=-1$ , and passes through the point $(3,5)$ .

I.

Write down the values of $h$ and $k$ .

[2]

II.

Find the value of $a$ , and hence write an expression for $g(x)$ .

[2]

Describe fully the transformations that transform the graph of $y=\frac{1}{x}$ to the graph of $y=g(x)$ .

[2]

Find the coordinates of the points where the graph of $y=g(x)$ intersects the line $y=2x-5$ .

[3]

Question 24

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=2^x$ , for $x\in\mathbb{R}$ . A function $g$ is defined by $g(x)=\frac{1}{2}f(x+2)-3$ .

I.

Describe fully the transformations that transform the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

Find the equation of the horizontal asymptote and the $y$ -intercept of the graph of $y=g(x)$ .

[2]

Solve $g(x)=13$ .

[2]

Find $g^{-1}(x)$ .

[1]

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Question 25

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=|x|$ , for $x\in\mathbb{R}$ . A function $g$ is defined by $g(x)=3-2|x-1|$ .

I.

Describe fully the transformations that transform the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

II.

Find the coordinates of the vertex and the intercepts with the coordinate axes.

[3]

Sketch the graph of $y=g(x)$ , clearly indicating the vertex and intercepts.

[2]

Solve $g(x)>0$ .

[1]

Question 26

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=x^2-6x+5$ , for $x\in\mathbb{R}$ , and let $g(x)=|f(x)|$ .

Write $f(x)$ in vertex form, and describe the transformations that transform the graph of $y=x^2$ to the graph of $y=f(x)$ .

[3]

Describe how the graph of $y=g(x)$ is obtained from the graph of $y=f(x)$ , and state the coordinates of its local minimum points.

[2]

Solve $g(x)=3$ .

[2]

Question 27

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A one-to-one function $f$ has domain $[-1,9]$ and range $[2,10]$ . The point $(7,3)$ lies on an extension of the graph of $y=f(x)$ .

A function $g$ is defined by

g(x)=5-f(2x+1)

Find the image of the point $(7,-3)$ on the graph of $y=g(x)$ .

[2]

Find the domain and range of $g$ .

[3]

Find an expression for $g^{-1}(x)$ in terms of $f^{-1}$ .

[2]

Question 28

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The polynomial function $f$ has zeros at $x=-1$ , $x=2$ and $x=5$ . A function $g$ is defined by

g(x)=-2f(3-x)+4

Find the values of $x$ for which $g(x)=4$ .

[3]

Describe the transformations that transform the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

A point on the graph of $y=f(x)$ has coordinates $(0,6)$ . Find the coordinates of its image on the graph of $y=g(x)$ .

[1]

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Question 29

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Let $f(x)=x^3-4x$ , for $x\in\mathbb{R}$ . A function $g$ is defined by
$g(x)=2f\left(\frac{x-4}{2}\right)-1$

Describe fully the transformations which map the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[4]

The graph of $y=f(x)$ has two stationary points.

I.

Find the coordinates of the two stationary points of $y=f(x)$ .

[2]

II.

Hence find the coordinates of the corresponding stationary points of $y=g(x)$ . If you did not obtain the stationary points in part (b)(i), use $(-1.15,3.08)$ and $(1.15,-3.08)$ .

[2]

Solve $g(x)=5$ .

[2]

Question 30

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A fountain jet is modelled by a transformed parabola $y=p(x)$ , where $x$ is the horizontal distance in metres and $y$ is the height in metres. The maximum height is $3.2$ m and occurs when $x=4$ . The jet passes through $(1,1.4)$ .

Find $p(x)$ in the form $p(x)=a(x-h)^2+k$ .

[3]

Consider the graph of $y=p(x)$ as a transformation of $y=x^2$ .

I.

Describe fully the transformations from $y=x^2$ to $y=p(x)$ .

[2]

II.

State the coordinates of the image of the vertex of $y=x^2$ .

[1]

Find the horizontal distance between the two points where the jet is at ground level.

[3]

Question 31

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The concentration $C$ of a medicine, in mg per litre, is modelled by
$C(t)=k+\frac{A}{t+2},\quad t\geq 0$
where $t$ is the time in hours. The horizontal asymptote is $C=1.5$ and the graph passes through $(1,4.5)$ .

Find the values of $k$ and $A$ .

[3]

The graph of $y=C(t)$ is obtained from the graph of $y=\frac{1}{x}$ .

I.

Describe fully the transformations.

[2]

II.

State the equations of the two asymptotes.

[1]

The medicine is considered active while $C(t)>2.2$ . Find how long after $t=0$ the medicine remains active.

[2]

Find the average concentration during the first $6$ hours.

[2]

Question 32

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Let $f(x)=e^x$ . The temperature $T$ , in degrees Celsius, of a cooling object is modelled by
$T(t)=18+14f(-0.35(t-2))$
where $t$ is measured in hours and $t\geq0$ .

Describe fully the transformations from the graph of $y=f(x)$ to the graph of $y=T(t)$ .

[4]

Use the model to answer the following.

I.

Find $T(0)$ .

[2]

II.

State the horizontal asymptote of the graph of $y=T(t)$ .

[1]

III.

Explain the meaning of this asymptote in the context of the model.

[1]

Find the time at which the temperature first reaches $22^\circ\text{C}$ .

[2]

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Question 33

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A function $f$ has domain $[-4,6]$ and range $[-2,5]$ . The zeros of $f$ are $x=-3$ , $x=1$ and $x=5$ . A function $g$ is defined by
$g(x)=3-f(2x-4)$

Describe fully the transformations which map the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

Determine the domain and range of $g$ .

I.

Find the domain of $g$ .

[2]

II.

Find the range of $g$ .

[1]

Solve $g(x)=3$ .

[2]

Question 34

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

f(x)=\ln(x^2+1),\quad x\in\mathbb{R}

For $a>0$ , define

g_a(x)=f(a(x-2))+1

It is given that $g_a(5)=3$ .

Find the value of $a$ .

[3]

Using this value of $a$ , describe the transformations from $y=f(x)$ to $y=g_a(x)$ .

[2]

Using the unrounded exact value of $a$ , solve $g_a(x)=2$ .

[2]

Question 35

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

For a function $f$ , two vertical transformations are considered: a vertical stretch by scale factor $p$ , where $p>0$ , and a vertical translation by $q$ units. The transformations are applied in different orders.

I.

Write an expression for the function obtained by applying the stretch first and then the translation.

[1]

II.

Write an expression for the function obtained by applying the translation first and then the stretch.

[1]

It is known that the graph obtained by translating first and then stretching is 10 units vertically above the graph obtained by stretching first and then translating. A point whose original $y$ -coordinate is $2$ has final $y$ -coordinate $11$ on the graph obtained by stretching first and then translating. Assume additionally that $1<p<3.5$ . Find $p$ and $q$ .

[3]

Prove that the two vertical transformations commute if and only if $p=1$ or $q=0$ .

[3]

Question 36

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Let $f(x)=\sin x$ , for $x\in\mathbb{R}$ . A function $g$ is defined by $g(x)=2-3\sin\left(\frac{x-\pi}{2}\right)$ , for $0\leq x\leq 5\pi$ .

I.

Describe fully the transformations that transform the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[4]

II.

State the period and range of $g$ .

[2]

I.

Solve $g(x)=2$ for $0\leq x\leq 5\pi$ .

[2]

II.

Find the values of $x$ in the interval $0\leq x\leq 5\pi$ at which $g$ attains its maximum and minimum values.

[2]

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Question 37

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A differentiable function $f$ has stationary points $A(-2,5)$ and $B(4,-1)$ , and a point of inflexion $C(1,2)$ . A transformed function is defined by $g(x)=a f(b(x-c))+d$ , where $a<0$ and $b>0$ . The images of $A$ and $B$ on the graph of $y=g(x)$ are $A'(-1,-6)$ and $B'(2,6)$ respectively.

I.

Find the values of $b$ and $c$ .

[4]

II.

Find the values of $a$ and $d$ .

[4]

Find the coordinates of the image of $C$ on the graph of $y=g(x)$ .

[1]

Assuming $f^{-1}$ exists on a suitable restricted domain, find $g^{-1}(x)$ in terms of $f^{-1}$ .

[1]

Question 38

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $f(x)=x^2-2x-3$ , for $x\in\mathbb{R}$ . A function $h$ is defined by $h(x)=|2f(x+1)+3|$ .

AI.

Write $f(x)$ in vertex form.

[2]

AII.

Let $k(x)=2f(x+1)+3$ . Find $k(x)$ and describe the transformations from $y=f(x)$ to $y=k(x)$ .

[3]

BI.

Find the zeros of $k$ .

[2]

BII.

Describe how the graph of $y=h(x)$ is obtained from the graph of $y=k(x)$ .

[1]

Solve $h(x)=3$ .

[2]

Question 39

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $f(x)=\arctan x$ , for $x\in\mathbb{R}$ . A function $g$ is defined by $g(x)=\frac{\pi}{6}-2\arctan(3x-6)$ .

I.

(a)(i) Describe fully the transformations that transform the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[4]

II.

(a)(ii) State the equations of the horizontal asymptotes of the graph of $y=g(x)$ .

[2]

I.

(b)(i) Find the $x$ -coordinate of the point where the graph of $y=g(x)$ crosses its midline.

[1]

II.

(b)(ii) Find $g^{-1}(x)$ and state its domain.

[3]

Question 40

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

The height $h$ metres of a cabin on a Ferris wheel is modelled by
$h(t)=a\cos(b(t-c))+d$
where $t$ is the time in minutes after the wheel starts moving. The maximum height is $28$ m, the minimum height is $4$ m, and one complete rotation takes $12$ minutes. The cabin is at maximum height when $t=2$ .

Find the values of $a$ , $b$ , $c$ and $d$ , taking $a>0$ , $b>0$ and $0\leq c<12$ .

[4]

Consider $h(t)$ as a transformation of $y=\cos x$ .

I.

Describe the horizontal transformations from $y=\cos t$ to the graph of $h(t)$ .

[2]

II.

Describe the vertical transformations from $y=\cos t$ to the graph of $h(t)$ .

[1]

During the first rotation, find the times when the cabin is $20$ m above the ground.

[2]

Find the length of time during the first rotation for which the cabin is above $20$ m.

[2]

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Question 41

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $f(x)=x^4-6x^2+2$ , for $x\in\mathbb{R}$ . For $a>0$ , define
$g_a(x)=f(a(x-1))+3$

Find the coordinates of the local maximum and local minimum points of $y=f(x)$ .

[3]

The horizontal distance between the two local minimum points of $y=g_a(x)$ is $\frac{4}{3}$ .

I.

Show that $a=\frac{3\sqrt3}{2}$ .

[2]

II.

Using this value of $a$ , find the local maximum point of $y=g_a(x)$ .

[1]

Using $a=\frac{3\sqrt3}{2}$ , solve $g_a(x)=0$ .

[4]

Describe fully the transformations from $y=f(x)$ to $y=g_a(x)$ when $a=\frac{3\sqrt3}{2}$ .

[2]

Question 42

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $f(x)=\arctan x$ , for $x\in\mathbb{R}$ . A function $g$ is defined by
$g(x)=3\arctan(2-x)-1$

Describe fully the transformations from the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[4]

Consider the asymptotes and intercept of $y=g(x)$ .

I.

State the equations of the two horizontal asymptotes.

[2]

II.

Find the $x$ -intercept of the graph of $y=g(x)$ .

[2]

Find the area between the graph of $y=g(x)$ and its lower horizontal asymptote for $2\leq x\leq6$ .

[3]

Question 43

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A function $f$ has points $A(-2,4)$ , $B(1,-1)$ and $C(5,7)$ on its graph. A transformed function is defined by
$g(x)=m f(k(x-a))+n$
where $m>0$ and $k>0$ . The corresponding image points on the graph of $y=g(x)$ are $A'(0,10)$ , $B'(1,0)$ and $C'\left(\frac{7}{3},16\right)$ .

Use the images of $A$ and $B$ to find $k$ and $a$ .

[3]

Use the images of $A$ and $B$ to find $m$ and $n$ .

[3]

Verify that $C$ is mapped to $C'$ .

I.

Find the image of the $x$ -coordinate of $C$ .

[1]

II.

Find the image of the $y$ -coordinate of $C$ .

[1]

Suppose $f$ is differentiable and has a stationary point at $x=r$ . Show that the corresponding stationary point of $g$ has $x$ -coordinate $\frac{r}{3}+\frac{2}{3}$ .

[3]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A function $f$ has three marked points $A(0,1)$ , $B(4,5)$ and $C(8,1)$ on its graph. The finite region between the graph of $y=f(x)$ , the $x$ -axis, and the lines $x=0$ and $x=8$ has area $\frac{88}{3}$ square units. A transformed function is defined by

g(x)=a f(b(x-c))+d

where $a>0$ and $b>0$ . The images of $A$ , $B$ and $C$ on the graph of $y=g(x)$ are $A'(2,0)$ , $B'(4,6)$ and $C'(6,0)$ respectively.

I.

Find the values of $b$ and $c$ .

[2]

II.

Find the values of $a$ and $d$ .

[2]

Find the area of the region between the graph of $y=g(x)$ , the $x$ -axis, and the lines $x=2$ and $x=6$ .

[3]

Suppose instead that a region under $y=f(x)$ from $x=r$ to $x=s$ has area $A$ and $f(x)\ge 0$ on this interval. For $G(x)=a f(b(x-c))+d$ , where $a>0$ and $b>0$ , show that the corresponding signed area is

\frac{aA+d(s-r)}{b}

[3]

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Question 45

HL • Paper 3

Hard

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HL • Paper 3

Hard

Calculator Permitted

The graph of $y=f(x)$ is transformed horizontally. A horizontal stretch by scale factor $s$ , where $s>0$ , and a horizontal translation by $a$ units are applied in different orders. A point $(u,v)$ on the original graph is followed through the transformations.

I.

Find the image of $(u,v)$ if the stretch is applied first and then the translation.

[1]

II.

Find the image of $(u,v)$ if the translation is applied first and then the stretch.

[1]

The graph of $y=f(x)$ has zeros at $x=-2$ and $x=6$ . A horizontal stretch by scale factor $\frac{3}{2}$ is applied first, followed by a translation $a$ units. The new zeros are at $x=-4$ and $x=8$ . Find $a$ .

[2]

Deduce the function notation for the transformed graph in part (b).

[1]

Show that a horizontal stretch by scale factor $s$ and a horizontal translation by $a$ commute if and only if $s=1$ or $a=0$ .

[3]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A parabolic arch is modelled as a transformation of the graph of $y=x^2$ . Its highest point is $(3,7)$ and it passes through $(1,3)$ . The model is written in the form

p(x)=k-a(x-h)^2

where $a>0$ .

I.

Find $h$ , $k$ and $a$ .

[3]

II.

Describe the transformations from $y=x^2$ to $y=p(x)$ .

[1]

Find the $x$ -intercepts of the arch, giving exact values.

[2]

A new family of arches is defined by $q_t(x)=t p(x)+4$ . Determine the values of $t$ , where $t$ is real, for which the graph of $y=q_t(x)$ has no $x$ -intercepts.

[4]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A transformed reciprocal graph has equation

g(x)=k+\frac{A}{x-h}, \quad x\ne h

Its vertical asymptote is $x=2$ , its horizontal asymptote is $y=-1$ , and it passes through $(3,5)$ .

I.

Find $h$ , $k$ and $A$ .

[3]

Describe the transformations from $y=\frac{1}{x}$ to $y=g(x)$ .

[2]

Find the exact area between the graph of $y=g(x)$ and its horizontal asymptote from $x=3$ to $x=5$ .

[2]

Show that for any point $(x,y)$ on the graph of $y=g(x)$ , the product of its horizontal and vertical directed distances from the asymptotes is constant.

[2]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A transformed logarithmic function has vertical asymptote $x=1$ and passes through the points $(2,0)$ and $(1+e^2,6)$ . It is assumed to have the form

g(x)=A\ln(x-C)+D

where $A>0$ .

I.

Find $A$ , $C$ and $D$ .

[3]

II.

Describe the transformations from $y=\ln x$ to $y=g(x)$ .

[1]

A second graph is defined by $h(x)=g(4-x)$ . State the domain and vertical asymptote of $h$ .

[2]

Find the point of intersection of the graphs of $y=g(x)$ and $y=h(x)$ .

[3]

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Question 49

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A sinusoidal graph is a transformation of $y=\sin x$ and has equation

g(x)=a\sin(b(x-c))+d

where $a>0$ and $b>0$ . Consecutive maximum points occur at $(1,5)$ and $(5,5)$ , and the minimum point between them is $(3,-1)$ .

I.

Find $a$ and $d$ .

[2]

II.

Find $b$ and one possible value of $c$ with $0\le c<4$ .

[3]

Using the values found in part (a), solve $g(x)=0$ for $0\le x\le 6$ .

[2]

The graph is transformed to $r(x)=g(x-m)+n$ . Determine $m$ and $n$ if the maximum point $(1,5)$ is mapped to $(7,2)$ .

[2]

Question 50

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $g(x)=\frac{2x+5}{x-1}$ , where $x\neq 1$ .

I.

Write $g(x)$ in the form $k+\frac{a}{x-h}$ , and state the equations of its asymptotes.

[2]

II.

Describe fully the transformations that transform the graph of $y=\frac{1}{x}$ to the graph of $y=g(x)$ .

[3]

Find $g^{-1}(x)$ .

[3]

Find the coordinates of the points where the graphs of $y=g(x)$ and $y=g^{-1}(x)$ intersect.

[2]

Question 51

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $f(x)=x^3-3x$ , for $x\in\mathbb{R}$ . A function $g$ is defined by $g(x)=2f\left(\frac{x+1}{2}\right)-1$ .

I.

Describe fully the transformations that transform the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[3]

II.

Show that $g'(x)=f'\left(\frac{x+1}{2}\right)$ .

[2]

Find the coordinates of the stationary points of the graph of $y=g(x)$ .

[3]

Solve $g(x)=-1$ .

[2]

Question 52

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A point $P(2,3)$ lies on the graph of $y=f(x)$ . The two sequences of transformations applied to the graph of $y=f(x)$ are:

Sequence A consists of a horizontal stretch by scale factor $2$ , followed by a translation $1$ unit to the right, followed by a vertical stretch by scale factor $3$ , followed by a translation $4$ units down.
Sequence B consists of a translation $1$ unit to the right, followed by a horizontal stretch by scale factor $2$ , followed by a translation $4$ units down, followed by a vertical stretch by scale factor $3$ .

Sequence A consists of a horizontal stretch by scale factor $2$ , followed by a translation $1$ unit to the right, followed by a vertical stretch by scale factor $3$ , followed by a translation $4$ units down.

I.

Write the transformed function in the form $y=Af(Bx+C)+D$ .

[2]

II.

Find the image of $P$ under Sequence A.

[2]

Sequence B consists of a translation $1$ unit to the right, followed by a horizontal stretch by scale factor $2$ , followed by a translation $4$ units down, followed by a vertical stretch by scale factor $3$ .

I.

Write the transformed function in the form $y=Af(Bx+C)+D$ .

[2]

II.

Find the image of $P$ under Sequence B.

[1]

A horizontal translation by $t$ units to the right, where $t\ge 0$ , and a horizontal stretch by scale factor $s$ , where $s>0$ , are applied in both possible orders. Determine the condition on $s$ and $t$ for the final graphs to be identical for every possible function $f$ . Justify your answer.

[3]

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Question 53

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $f(x)=x^3-3x$ , for $x\in\mathbb{R}$ . A function $g$ is defined by
$g(x)=\left|2f(x-1)-1\right|$

Describe how the graph of $y=g(x)$ is obtained from the graph of $y=f(x)$ .

[4]

The local minimum points of $y=g(x)$ occur where $2f(x-1)-1=0$ .

I.

Write down the equation that must be solved to find the $x$ -coordinates of the local minimum points of $y=g(x)$ .

[1]

II.

Find the coordinates of the local minimum points of $y=g(x)$ .

[2]

Solve $g(x)=4$ .

[5]

Question 54

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $f(x)=x+e^x$ , for $x\in\mathbb{R}$ . A function $g$ is defined by
$g(x)=4-2f(3-x)$

Describe fully the transformations from the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[4]

Since $f$ is one-to-one, $g^{-1}$ exists.

I.

Find an expression for $g^{-1}(x)$ in terms of $f^{-1}$ .

[2]

II.

Find $g^{-1}(0)$ .

[1]

Find the fixed point of $g$ , that is, solve $g(x)=x$ .

[3]

Question 55

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For a real parameter $a$ , define
$g_a(x)=a+\frac{4}{x-a},\quad x\ne a$

Describe the transformations from the graph of $y=\frac{1}{x}$ to the graph of $y=g_a(x)$ .

[3]

Consider the asymptotes of $y=g_a(x)$ .

I.

State the equations of the asymptotes.

[1]

II.

Hence describe the locus of the point of intersection of the asymptotes as $a$ varies.

[1]

Find the coordinates of the two points of intersection of the graph of $y=g_a(x)$ with the line $y=x$ .

[3]

Prove that the distance between these two intersection points is independent of $a$ , and state this distance.

[2]

Question 56

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let

f(x)=(x-2)^2-1, \quad x\in\mathbb{R}

A transformed modulus function is defined by

g(x)=\left|2f(x-1)-4\right|

I.

Write $g(x)$ in the form $|A(x-h)^2+B|$ .

[2]

II.

Describe the transformation represented by replacing $2f(x-1)-4$ with its modulus.

[1]

Find the coordinates of all local minimum points of $y=g(x)$ .

[3]

Determine the range of $g$ . Justify your answer using transformations.

[3]

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Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let $f$ be a one-to-one function with inverse $f^{-1}$ . A transformed function is defined by

g(x)=3f(2x-4)+1

I.

Describe fully the transformations from $y=f(x)$ to $y=g(x)$ .

[2]

II.

A point $(10,-2)$ lies on $y=f(x)$ . Find the corresponding point on $y=g(x)$ .

[1]

Find an expression for $g^{-1}(x)$ in terms of $f^{-1}$ .

[3]

Show that the image of $(7,-5)$ under reflection in the line $y=x$ lies on the graph of $y=g^{-1}(x)$ .

[2]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A transformation $T$ maps a point $(x,y)$ on the graph of a function to the point

T(x,y)=\left(\frac{x}{2}+4,\,-2y+3\right)

Repeated applications of $T$ are considered, starting from a point $P_0=(0,0)$ on an initial graph. Let $P_n=(x_n,y_n)$ after $n$ applications of $T$ .

I.

Find $P_1$ and $P_2$ .

[2]

II.

Write down recurrence relations for $x_n$ and $y_n$ .

[1]

Find the fixed point of $T$ .

[2]

Show that $x_n=8\left(1-\left(\frac{1}{2}\right)^n\right)$ .

[2]

Determine whether the sequence of points $P_n$ approaches the fixed point. Justify your answer.

[3]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For each real value of $t$ , define

g_t(x)=t+\frac{1}{x-t},\quad x\ne t

Each graph is obtained from the graph of $y=\frac{1}{x}$ by a transformation.

I.

Describe the transformation from $y=\frac{1}{x}$ to $y=g_t(x)$ .

[1]

II.

State the equations of the asymptotes of $y=g_t(x)$ .

[2]

Find the intersections of $y=g_t(x)$ with the line $y=x$ .

[3]

Show that the distance between the two intersection points found in part (b) is independent of $t$ .

[2]

Deduce the set of centres of the family of graphs $y=g_t(x)$ .

[1]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The function

f(x)=e^{-x^2},\quad x\in\mathbb{R}

has a bell-shaped graph with maximum value $1$ . A transformed family is defined by

g(x)=a f(bx)+1

where $a>0$ and $b>0$ . The width of the graph of $y=g(x)$ at height $1+\frac{a}{2}$ is the horizontal distance between the two points where $g(x)=1+\frac{a}{2}$ .

I.

Show that the two $x$ -coordinates where $g(x)=1+\frac{a}{2}$ are $x=\pm\frac{\sqrt{\ln 2}}{b}$ .

[3]

II.

Hence write the width at height $1+\frac{a}{2}$ in terms of $b$ .

[1]

A particular graph in the family has maximum value $5$ and width $3$ at half its height above the horizontal asymptote. Find $a$ and $b$ .

[3]

It is known that

\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}

For the graph found in part (b), find the exact area between $y=g(x)$ and its horizontal asymptote $y=1$ over all real $x$ .

[3]

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