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

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

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

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

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

Translations of Graphs (DP IB Maths: AI HL)