Name: Translations of Graphs (2.5.1) | DP IB Maths: AI HL Revision Notes 2021 Membership
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2.5.1 Translations of Graphs

Translations of Graphs

What are translations of graphs?

When you alter a function in certain ways, the effects on the graph of the function can be described by geometrical transformations
For a translation:
- the graph is moved (up or down, left or right) in the xy plane
  - Its position changes
- the shape, size, and orientation of the graph remain unchanged
A particular translation (how far left/right, how far up/down) is specified by a translation vector $(\begin{matrix} x \\ y \end{matrix})$ :
- x is the horizontal displacement
  - Positive moves right
  - Negative moves left
- y is the vertical displacement
  - Positive moves up
  - Negative moves down

What effects do horizontal translations have on the graphs and functions?

A horizontal translation of the graph $y = f (x)$ by the vector $(\begin{matrix} a \\ 0 \end{matrix})$ is represented by
- $y = f (x - a)$
The x-coordinates change
- The value a is subtracted from them
The y-coordinates stay the same
The coordinates $(x, y)$ become $(x - a, y)$
Horizontal asymptotes stay the same
Vertical asymptotes change
- $x = k$ becomes $x = k - a$

Translations statement_horiz_Illustration

What effects do vertical translations have on the graphs and functions?

A vertical translation of the graph $y = f (x)$ by the vector $(\begin{matrix} 0 \\ b \end{matrix})$ is represented by
- $y - b = f (x)$
- This is often rearranged to $y = f (x) + b$
The x-coordinates stay the same
The y-coordinates change
- The value b is added to them
The coordinates $(x, y)$ become $(x, y + b)$
Horizontal asymptotes change
- $y = k$ becomes $y = k + b$
Vertical asymptotes stay the same

Exam Tip

To get full marks in an exam make sure you use correct mathematical terminology
- For example: Translate by the vector $(\begin{matrix} 2 \\ - 4 \end{matrix})$

Worked Example

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

we-image

Sketch the graph of

y = f (x + 3)

2-5-1-ib-aa-sl-translate-graph-a-we-solution

Sketch the graph of

y = f (x) + 3

2-5-1-ib-aa-sl-translate-graph-b-we-solution

DP IB Maths: AI HL

Revision Notes

2.5.1 Translations of Graphs

Translations of Graphs

What are translations of graphs?

What effects do horizontal translations have on the graphs and functions?

What effects do vertical translations have on the graphs and functions?

Exam Tip

Worked Example

Author: Daniel

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