Composite Transformations of Graphs (DP IB Maths: AI HL)

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Dan

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2 October 2023

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Composite Transformations of Graphs

What transformations do I need to know?

$y = f (x + k)$ is horizontal translation by vector $(\begin{matrix} - k \\ 0 \end{matrix})$
- If k is positive then the graph moves left
- If k is negative then the graph moves right
$y = f (x) + k$ is vertical translation by vector $(\begin{matrix} 0 \\ k \end{matrix})$
- If k is positive then the graph moves up
- If k is negative then the graph moves down
$y = f (k x)$ is a horizontal stretch by scale factor $\frac{1}{k}$ centred about the y-axis
- If k > 1 then the graph gets closer to the y-axis
- If 0 < k < 1 then the graph gets further from the y-axis
$y = k f (x)$ is a vertical stretch by scale factor $k$ centred about the x-axis
- If k > 1 then the graph gets further from the x-axis
- If 0 < k < 1 then the graph gets closer to the x-axis
$y = f (- x)$ is a horizontal reflection about the y-axis
- A horizontal reflection can be viewed as a special case of a horizontal stretch
$y = - f (x)$ is a vertical reflection about the x-axis
- A vertical reflection can be viewed as a special case of a vertical stretch

How do horizontal and vertical transformations affect each other?

Horizontal and vertical transformations are independent of each other
- The horizontal transformations involved will need to be applied in their correct order
- The vertical transformations involved will need to be applied in their correct order
Suppose there are two horizontal transformation H₁then H₂ and two vertical transformations V₁then V₂thenthey can be applied in the following orders:
- Horizontal then vertical:
  - H₁H₂V₁V₂
- Vertical then horizontal:
  - V₁V₂H₁H₂
- Mixed up (provided that H₁ comes before H₂ and V₁ comes before V2):
  - H₁V₁H₂V₂
  - H₁V₁V₂H₂
  - V₁H₁V₂H₂
  - V₁H₁H₂V₂

Examiner Tip

In an exam you are more likely to get the correct solution if you deal with one transformation at a time and sketch the graph after each transformation

Worked example

The diagram below shows the graph of $y = f (x)$ .

we-image

Sketch the graph of $y = \frac{1}{2} f (\frac{x}{2})$ .

2-5-4-ib-aa-sl-comp-transformation-a-we-solution

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Composite Vertical Transformations af(x)+b

How do I deal with multiple vertical transformations?

Order matters when you have more than one vertical transformations
If you are asked to find the equation then build up the equation by looking at the transformations in order
- A vertical stretch by scale factor a followed by a translation of $(\begin{matrix} 0 \\ b \end{matrix})$
  - Stretch: $y = a f (x)$
  - Then translation: $y = [a f (x)] + b$
  - Final equation: $y = a f (x) + b$
- A translation of $(\begin{matrix} 0 \\ b \end{matrix})$ followed by a vertical stretch by scale factor a
  - Translation: $y = f (x) + b$
  - Then stretch: $y = a [f (x) + b]$
  - Final equation: $y = a f (x) + a b$
If you are asked to determine the order
- The order of vertical transformations follows the order of operations
- First write the equation in the form $y = a f (x) + b$
  - First stretch vertically by scale factor a
  - If a is negative then the reflection and stretch can be done in any order
  - Then translate by $(\begin{matrix} 0 \\ b \end{matrix})$

Worked example

The diagram below shows the graph of $y = f (x)$ .

we-image

Sketch the graph of $y = 3 f (x) - 2$ .

2-5-4-ib-aa-sl-comp-transformation-b-we-solution

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Composite Horizontal Transformations f(ax+b)

How do I deal with multiple horizontal transformations?

Order matters when you have more than one horizontal transformations
If you are asked to find the equation then build up the equation by looking at the transformations in order
- A horizontal stretch by scale factor $\frac{1}{a}$ followed by a translation of $(\begin{matrix} - b \\ 0 \end{matrix})$
  - Stretch: $y = f (a x)$
  - Then translation: $y = f (a (x + b))$
  - Final equation: $y = f (a x + a b)$
- A translation of $(\begin{matrix} - b \\ 0 \end{matrix})$ followed by a horizontal stretch by scale factor $\frac{1}{a}$
  - Translation: $y = f (x + b)$
  - Then stretch: $y = f ((a x) + b)$
  - Final equation: $y = f (a x + b)$
If you are asked to determine the order
- First write the equation in the form $y = f (a x + b)$
- The order of horizontal transformations is the reverse of the order of operations
  - First translate by $(\begin{matrix} - b \\ 0 \end{matrix})$
  - Then stretch by scale factor $\frac{1}{a}$
  - If a is negative then the reflection and stretch can be done in any order

Worked example

The diagram below shows the graph of $y = f (x)$ .

we-image Sketch the graph of $y = f (2 x - 1)$ .

2-6-4-ib-aa--ai-hl-comp-horizontal-trans-we-solution

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Composite Transformations of Graphs (DP IB Maths: AI HL)

Revision Note

What transformations do I need to know?

How do horizontal and vertical transformations affect each other?

How do I deal with multiple vertical transformations?

How do I deal with multiple horizontal transformations?

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Author: Dan