DP Maths AI HL 2.9 Transformations of Functions

DP Maths AI HL 2.9 Transformations of Functions - Full notes

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DP Maths AI HL | 2.9 Transformations of Functions

DP MATHEMATICS AI · TOPIC 2: FUNCTIONS

2.9

Transformations of Functions

Full notes for Applications and Interpretation HL

Contents

1	What is a transformation?
2	Vertical translations
3	Horizontal translations
4	Reflections in the x-axis
5	Reflections in the y-axis
6	Vertical stretches
7	Horizontal stretches
8	Combining transformations: order matters
9	Transforming points and key features
10	Transforming asymptotes
11	Transformations of quadratic functions
12	Transformations of exponential functions
13	Transformations of sine and cosine
14	Finding equations from graphs
15	Transformations in modelling
16	Describing transformations precisely
17	Exploring transformations with technology
18	Mixed review: worked problems
19	Transformations of logarithmic and rational functions (HL)
20	Transformations and inverse functions (HL)
21	Investigation: modelling daylight hours
22	Exam-style questions with markscheme
23	Common misconceptions and links
24	Check your understanding
25	Summary and topic map
26	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the practice exercise.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
apply and describe vertical and horizontal translations
apply and describe reflections in the axes
apply and describe vertical and horizontal stretches
combine transformations in the correct order
find images of points, key features and asymptotes
transform quadratic, exponential and sinusoidal functions
find equations of transformed graphs
use transformations to adapt models

HL booklet: this booklet contains all the SL content plus transformations of logarithmic and rational functions and the link with inverse functions.

Guiding questions

How do changes to a function's equation move or reshape its graph? How can a model be adapted to new data by transforming it?

How this topic is examined

Transformations are tested in short Paper 1 questions (describe a transformation, find the image of a point, write the new equation) and support modelling questions in Paper 2, especially for sinusoidal and exponential models.

Use precise language: 'translation by the vector (a, b)', 'vertical stretch with scale factor p', 'horizontal stretch with scale factor 1/q', 'reflection in the x-axis'.

Mathematics: Applications and Interpretation places strong emphasis on technology, modelling and interpretation. Most papers allow a GDC, so you are expected to use it efficiently, record the method you used, and interpret every answer in context.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working, including what you entered into a GDC
Show that	obtain the given result; every step must be shown and the result must not be assumed
Estimate	obtain an approximate value
Sketch	a diagram showing the general shape and key features, labelled
Interpret / Comment	explain the meaning of a result in context, or give a judgement
Justify / Explain	give valid reasons or evidence
Hence	use the previous result

How to use these notes

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or explanation of a key result
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given), graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet

1 What is a transformation?

Moving and reshaping graphs.

Think about it

The graph of y = x2 has its vertex at the origin. Predict where the vertex of y = (x − 3)2 + 2 is. Then check with a GDC.

A transformation changes the position or shape of a graph. Translations move it; reflections flip it; stretches scale it. Knowing how each change to the equation affects the graph lets you sketch complicated functions from simple ones, and adapt models to fit new data.

There are two kinds of change: outside the function (changes to f(x) itself, which act vertically on y-values) and inside the function (changes to x before applying f, which act horizontally). Inside changes act in the 'opposite' way to what you might expect.

Equation	Transformation
y = f(x) + b	vertical translation by b
y = f(x − a)	horizontal translation by a
y = −f(x)	reflection in the x-axis
y = f(−x)	reflection in the y-axis
y = pf(x)	vertical stretch, scale factor p
y = f(qx)	horizontal stretch, scale factor 1/q

2 Vertical translations

Shift up or down.

y = f(x) + b moves every point up by b (down if b < 0): (x, y) → (x, y + b). The shape is unchanged.

Figure 1. Vertical translations of y = x2.

WORKED EXAMPLE 1 Vertical translation (basic)

Describe the transformation from y = 2x to y = 2x − 5 and state the new asymptote.

Solution

Translation 5 units downA1

Asymptote y = −5A1

Practice exercise 1

1. Write the equation after y = √x is translated 4 units up.

2. Find the image of (2, 7) under y = f(x) − 3.

3 Horizontal translations

Shift left or right.

y = f(x − a) moves every point a units to the right (left if a < 0): (x, y) → (x + a, y). The sign is opposite to what appears in the bracket because the new function reaches the same output when x − a equals the old input, that is, a units later.

Figure 2. Horizontal translations of y = x2.

WORKED EXAMPLE 2 Horizontal translation (standard)

Describe y = (x + 4)3 as a transformation of y = x3.

Solution

x replaced by x + 4 = x − (−4)M1

Translation 4 units leftA1

Common mistake

Saying y = f(x + 4) moves the graph 4 units right. It moves 4 units left.

Practice exercise 2

1. Write the equation after y = ex is translated 3 units right.

2. Describe y = √(x − 1) + 2 as a transformation of y = √x.

3. Find the image of (−1, 5) under y = f(x + 2).

4 Reflections in the x-axis

Flip vertically.

y = −f(x) changes the sign of every y-value: (x, y) → (x, −y). Roots stay fixed; maxima become minima.

Figure 3. Reflection in the x-axis.

Practice exercise 3

1. Find the image of (3, −4) under y = −f(x).

2. Describe the graph of y = −x2 + 1 as transformations of y = x2.

5 Reflections in the y-axis

Flip horizontally.

y = f(−x) changes the sign of every x-value: (x, y) → (−x, y). The y-intercept stays fixed. Even functions (such as x2 and cos x) are unchanged by this reflection.

Figure 4. Reflection in the y-axis turns growth into decay.

Practice exercise 4

1. Write the equation after y = √x is reflected in the y-axis and state its domain.

2. Find the image of (−2, 6) under y = f(−x).

6 Vertical stretches

Scale the outputs.

y = pf(x) multiplies every y-value by p: (x, y) → (x, py). Roots stay fixed. If 0 < p < 1 the graph is compressed towards the x-axis. For sinusoidal functions this changes the amplitude.

Figure 5. Vertical stretches of y = sin x.

Practice exercise 5

1. Find the image of (4, −6) under y = 0.5f(x).

2. y = f(x) has a maximum at (2, 5). Find the maximum of y = 3f(x).

7 Horizontal stretches

Scale the inputs.

y = f(qx) divides every x-value by q: (x, y) → (x/q, y). This is a horizontal stretch with scale factor 1/q. If q > 1 the graph is compressed towards the y-axis. For sinusoidal functions this changes the period: y = sin(qx) has period 2π/q (or 360°/q).

Figure 6. y = sin 2x has half the period of y = sin x.

Common mistake

Calling y = f(3x) a horizontal stretch with scale factor 3. The scale factor is 1/3.

Practice exercise 6

1. Find the image of (6, 2) under y = f(2x).

2. State the period of y = cos(x/3).

3. The roots of f are −4 and 8. Find the roots of y = f(4x).

8 Combining transformations: order matters

Vertical and horizontal separately.

Treat vertical changes (outside) and horizontal changes (inside) separately. Vertical: for y = pf(x) + b, stretch by p first, then translate by b. Horizontal: for y = f(q(x − a)), stretch by 1/q, then translate by a; if the equation is given as f(qx − c), factorize first: f(q(x − c/q)).

Horizontal and vertical transformations do not interfere with each other, so their relative order does not matter.

WORKED EXAMPLE 3 Describing a sequence (standard)

Describe the transformations from y = f(x) to y = 2f(3x − 6) + 1.

Solution

y = 2f(3(x − 2)) + 1M1

Horizontal stretch factor 1/3, then translation 2 rightA1

Vertical stretch factor 2, then translation 1 upA1

WORKED EXAMPLE 4 Order changes the result (exam level)

y = f(x) is translated 3 up, then stretched vertically by factor 2. Compare with the reverse order.

Solution

Translate then stretch: 2(f(x) + 3) = 2f(x) + 6A1

Stretch then translate: 2f(x) + 3A1

Different, so order mattersR1

Practice exercise 7

1. Write the equation when y = f(x) is reflected in the x-axis then translated 2 down.

2. Describe y = 4 − f(x + 1).

3. Describe y = f(2x + 8).

9 Transforming points and key features

Follow the coordinates.

To find the image of a point, apply the horizontal changes to x and the vertical changes to y, in order. Key features (intercepts, maxima, minima) move in the same way, which allows quick sketches.

WORKED EXAMPLE 5 Image of a point (standard)

(4, 6) lies on y = f(x). Find its image on y = 3f(x − 2) − 5 and on y = f(½x).

Solution

(4 + 2, 3 × 6 − 5) = (6, 13)A1

(4 × 2, 6) = (8, 6)A1

Practice exercise 8

1. (−3, 2) lies on y = f(x). Find its image on y = −2f(x) + 1.

2. A maximum at (1, 4) on f. Find the corresponding point on y = f(x − 3) − 4 and state its nature.

10 Transforming asymptotes

Lines move too.

Vertical asymptotes x = k move like x-coordinates; horizontal asymptotes y = k move like y-coordinates. For example, y = 1/x has asymptotes x = 0 and y = 0, so y = 3/(x − 2) + 1 has asymptotes x = 2 and y = 1.

Figure 7. Asymptotes move with the graph.

Practice exercise 9

1. State the asymptotes of y = 2x+1 − 3.

2. y = f(x) has asymptotes x = −1 and y = 4. Find the asymptotes of y = ½f(x − 3).

11 Transformations of quadratic functions

Vertex form.

y = a(x − h)2 + k is y = x2 stretched vertically by a (reflected if a < 0) and translated by (h, k). The vertex moves from (0, 0) to (h, k).

WORKED EXAMPLE 6 Vertex form (standard)

Describe y = −2(x + 1)2 + 8 as transformations of y = x2 and state the vertex.

Solution

Vertical stretch factor 2 and reflection in x-axis; translation by (−1, 8)A1 A1

Vertex (−1, 8), a maximumA1

Practice exercise 10

1. Write the equation of y = x2 after a vertical stretch by 3 and translation by (2, −5).

2. Find the vertex and range of y = 0.5(x − 4)2 + 1.

12 Transformations of exponential functions

Shifts change the asymptote.

y = kax−h + c is y = ax stretched by k, translated h right and c up. The asymptote y = 0 moves to y = c. A horizontal translation is equivalent to a vertical stretch for exponentials: ax−h = a−hax.

WORKED EXAMPLE 7 Exponential transformation (standard)

Describe y = 5 − 3e−0.5x as transformations of y = ex.

Solution

Horizontal stretch factor 2 and reflection in the y-axis (e−0.5x)A1

Vertical stretch 3 and reflection in x-axis, then translation 5 up; asymptote y = 5A1 A1

Practice exercise 11

1. Show that 2x−3 is a vertical stretch of 2x and find the factor.

2. Write y = ex after a reflection in the y-axis and translation by (0, 2).

13 Transformations of sine and cosine

Amplitude, period, phase, axis.

|a| is the amplitude (vertical stretch), 2π/b (or 360°/b) the period (horizontal stretch), c the phase shift (horizontal translation) and y = d the principal axis (vertical translation).

Figure 8. Amplitude 2, period π, shift π/4 right, axis y = 1.

Practice exercise 12

1. State the amplitude, period and principal axis of y = 4 cos(0.5x) − 1.

2. Write the equation after y = sin x is stretched vertically by 3 and horizontally by factor ½.

14 Finding equations from graphs

Work backwards.

WORKED EXAMPLE 8 From a graph (exam level)

y = a(b)x + c has asymptote y = 4, passes through (0, 10) and (1, 7). Find a, b and c.

Solution

c = 4; a + 4 = 10, a = 6A1 A1

6b + 4 = 7, b = 0.5A1

WORKED EXAMPLE 9 Parabola from vertex (standard)

A parabola has vertex (3, −2) and passes through (4, 0). Find its equation.

Solution

y = a(x − 3)2 − 2M1

0 = a − 2, a = 2: y = 2(x − 3)2 − 2A1

Practice exercise 13

1. A sine curve has maximum 9, minimum 1 and period 12, with a maximum at x = 3. Write it as y = a cos(b(x − c)) + d.

2. Find c if y = ln(x − c) has its x-intercept at x = 5.

15 Transformations in modelling

Adapting a model to new conditions.

Transformations adapt models: changing units (a vertical or horizontal stretch), changing the starting time (a horizontal translation), or adding a fixed amount (a vertical translation). For example, if T(t) gives a temperature in °C, then 1.8T(t) + 32 gives it in °F; T(t − 3) is the same pattern 3 hours later.

WORKED EXAMPLE 10 Changing a model (exam level)

The height of a tide is h(t) = 3 sin(0.5t) + 5 metres, t hours after midnight. Write the model for a port where the tide is 2 hours later and 1 m higher at every time.

Solution

Shift 2 right: 3 sin(0.5(t − 2)) + 5A1

Then 1 up: h(t) = 3 sin(0.5(t − 2)) + 6A1

Practice exercise 14

1. A model gives cost C(n) in USD. Write the cost in INR if 1 USD = 83 INR.

2. P(t) is a population t years after 2000. Write the model with t measured from 2010.

16 Describing transformations precisely

Language that earns marks.

Say	Not
translation by the vector (2, −3)	moved 2 and −3
vertical stretch, scale factor 3	made bigger by 3
horizontal stretch, scale factor 1/2	squashed by 2
reflection in the x-axis	flipped upside down

17 Exploring transformations with technology

Sliders and parameters.

GDC tip

Graph y = a(x − h)2 + k with sliders for a, h and k (many GDCs and software support this) and observe each parameter's effect. Then match a curve to data by adjusting the parameters until it fits, before using regression.

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 11 Mixed problem 1 (exam level)

f(x) = x2. The graph is translated by (−2, 3) and reflected in the x-axis (in that order). Find the new equation and vertex.

Solution

Translation: (x + 2)2 + 3A1

Reflection: y = −(x + 2)2 − 3, vertex (−2, −3)A1 A1

Practice exercise 15

1. Describe y = √(4x) in two different ways.

2. (5, −1) lies on f. Find its image on y = 2 − 3f(x/2).

19 Transformations of logarithmic and rational functions (HL)

Asymptotes and intercepts.

y = a ln(b(x − h)) + k has vertical asymptote x = h. y = a/(x − h) + k has asymptotes x = h and y = k. Identify the parent function, then apply the transformations to its key features.

WORKED EXAMPLE 12 Log transformation (standard)

Describe y = 2 ln(x − 3) + 1 and find its asymptote and x-intercept.

Solution

Translation 3 right, vertical stretch 2, translation 1 upA1

Asymptote x = 3; ln(x − 3) = −½ gives x = 3 + e−0.5 = 3.61A1 A1

Practice exercise 16

1. Find the asymptotes of y = −4/(x + 1) + 2.

2. Write y = ln x after a horizontal stretch factor 3 and translation 2 down.

20 Transformations and inverse functions (HL)

Reflection in y = x.

The inverse of a transformed function undoes the transformations in reverse order. If f(x) = 2x + 3 (stretch then translate), then f−1(x) = (x − 3)/2 (translate back, then stretch back). Graphically the inverse is the reflection in y = x, which swaps horizontal and vertical transformations.

Practice exercise 17

1. Find the inverse of f(x) = 3ex − 1 by undoing the transformations.

2. If f has asymptote y = 2, state the asymptote of f−1.

21 Investigation: modelling daylight hours

A Paper 3 style exploration.

Daylight in a city varies from 9.2 hours (21 December, day 355) to 15.0 hours (21 June, day 172). 1. Find the amplitude and principal axis of a sinusoidal model. 2. Explain why the period is 365 days and find b. 3. Write a model D(t) = a cos(b(t − c)) + d with a maximum at t = 172. 4. Predict the daylight on 1 March (t = 60). 5. A city further north has a range of 6.5 to 17.7 hours. Describe the transformation needed to adapt your model.

Answers. 1. a = 2.9, d = 12.1. 2. one cycle per year: b = 2π/365. 3. D = 2.9 cos(2π(t − 172)/365) + 12.1. 4. D(60) ≈ 11.1 hours. 5. A vertical stretch (amplitude 5.6) and a vertical translation (axis 12.1, unchanged here).

22 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 10 minutes)

The graph of y = f(x) passes through A(2, −3) and has a horizontal asymptote y = 1. g(x) = 2f(x + 1) − 4.

(a) Describe the transformations from f to g. [3]

(b) Find the image of A. [2]

Markscheme

Part	Answer	Marks
(a)	translation 1 left	A1
	vertical stretch factor 2	A1
	translation 4 down (after the stretch)	A1
(b)	(1, −10)	A2
(c)	y = −2	A1

Examiner's comments. Stretch before translate for vertical changes.

Question 2 (Paper 2 style) (about 12 minutes)

A water wave has height h(t) = 0.8 sin(2t) metres.

(a) State the amplitude and period. [2]

(b) A second wave has twice the amplitude, half the period, and starts 0.5 s later. Write its equation. [3]

Markscheme

Part	Answer	Marks
(a)	0.8 m	A1
	π s	A1
(b)	amplitude 1.6	A1
	b = 4	A1
	h2 = 1.6 sin(4(t − 0.5))	A1
(c)	4(t − 0.5) = π/2	M1
	t = 0.893 s	A1

Examiner's comments. The phase shift must be included when locating the maximum.

Question 3 (Paper 1 style) (about 8 minutes)

The graph of y = x2 is stretched vertically by factor 3, then translated by (1, −4). Find the equation and the x-intercepts. [4]

Markscheme

Part	Answer	Marks
	y = 3(x − 1)2 − 4	A2
	(x − 1)2 = 4/3	M1
	x = 1 ± 1.15: x = −0.155, 2.15	A1

Examiner's comments. Apply the stretch before the translation.

23 Common misconceptions and links

Students often think…	Actually…
f(x − 3) moves left	it moves right
f(2x) stretches by 2	by 1/2
order never matters	stretch before translate in each direction
asymptotes stay fixed	they move with the graph

How 2.9 connects to the rest of the course

Topic	Connection
2.5 Polynomials	vertex form
3.7 / 2.5 Trigonometric functions	amplitude and period
1.5 Exponential functions	asymptotes
2.2 Functions	inverse functions

24 Check your understanding

Section A: multiple choice

1. y = f(x) + 5 is

A. a translation 5 up B. 5 right C. a stretch D. 5 left

2. y = f(x/2) has horizontal scale factor

A. 1/2 B. −2 C. 2 D. 4

3. y = −f(x) is a reflection in

A. the y-axis B. the x-axis C. y = x D. the origin

4. The period of sin 4x is

A. 4π B. 2π C. π/4 D. π/2

5. y = 3 − 2x has asymptote

A. y = 0 B. y = 3 C. y = −3 D. x = 3

Section B: structured

6. (2, 8) lies on f. Find its image on y = f(−x) + 1. [2]

7. Describe y = 5(x + 3)2 as transformations of y = x2. [2]

Answers

1. A 2. C 3. B 4. D 5. B

6. (−2, 9)

7. vertical stretch factor 5; translation 3 left

25 Summary and topic map

Idea	Remember
Translations	f(x) + b; f(x − a): vector (a, b)
Reflections	−f(x): x-axis; f(−x): y-axis
Stretches	pf(x): vertical p; f(qx): horizontal 1/q
Order	stretch/reflect before translate in each direction
Sinusoidal	amplitude, period, phase shift, axis
Modelling	unit changes and time shifts

Before the exam, make sure you…

•use precise language
•factorize inside brackets
•move asymptotes too
•check with a point

Transformations of Functions on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 9. Topic map for Transformations of Functions.

26 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. y = √x + 4

2. (2, 4)

Practice exercise 2

1. y = ex−3

2. translation by the vector (1, 2)

3. (−3, 5)

Practice exercise 3

1. (3, 4)

2. reflection in the x-axis then translation 1 up

Practice exercise 4

1. y = √(−x), x ≤ 0

2. (2, 6)

Practice exercise 5

1. (4, −3)

2. (2, 15)

Practice exercise 6

1. (3, 2)

2. 6π

3. −1 and 2

Practice exercise 7

1. y = −f(x) − 2

2. translation 1 left; reflection in x-axis; translation 4 up

3. f(2(x + 4)): horizontal stretch factor ½, then 4 left

Practice exercise 8

1. (−3, −3)

2. (4, 0): maximum

Practice exercise 9

1. y = −3

2. x = 2 and y = 2

Practice exercise 10

1. y = 3(x − 2)2 − 5

2. (4, 1); y ≥ 1

Practice exercise 11

1. 2x−3 = (1/8)2x: factor 1/8

2. y = e−x + 2

Practice exercise 12

1. 4; 4π; y = −1

2. y = 3 sin 2x

Practice exercise 13

1. y = 4 cos(π(x − 3)/6) + 5

2. c = 4

Practice exercise 14

1. 83C(n)

2. P(t + 10)

Practice exercise 15

1. horizontal stretch factor 1/4, or vertical stretch factor 2 (since √(4x) = 2√x)

2. (10, 5)

Practice exercise 16

1. x = −1, y = 2

2. y = ln(x/3) − 2

Practice exercise 17

1. ln((x + 1)/3)

2. x = 2

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