Stretches 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 stretch:
- the graph is stretched about one of the coordinate axes by a scale factor
  - Its size changes
- the orientation of the graph remains unchanged
A particular stretch is specified by a coordinate axis and a scale factor:
- The distance between a point on the graph and the specified coordinate axis is multiplied by the constant scale factor
- The graph is stretched in the direction which is parallel to the other coordinate axis
- For scale factors bigger than 1
  - the points on the graph get further away from the specified coordinate axis
- For scale factors between 0 and 1
  - the points on the graph get closer to the specified coordinate axis
  - This is also sometimes called a compression but in your exam you must use the term stretch with the appropriate scale factor

A horizontal stretch of the graph $y = f (x)$ by a scale factor q centred about the y-axis is represented by
- $y = f (\frac{x}{q})$
The x-coordinates change
- They are divided by q
The y-coordinates stay the same
The coordinates $(x, y)$ become $(q x, y)$
Horizontal asymptotes stay the same
Vertical asymptotes change
- $x = k$ becomes $x = q k$

Stretches statement_horiz_Illustration

A vertical stretch of the graph $y = f (x)$ by a scale factor p centred about the x-axis is represented by
- $\frac{y}{p} = f (x)$
- This is often rearranged to $y = p f (x)$
The x-coordinates stay the same
The y-coordinates change
- They are multiplied by p
The coordinates $(x, y)$ become $(x, p y)$
Horizontal asymptotes change
- $y = k$ becomes $y = p k$
Vertical asymptotes stay the same

To get full marks in an exam make sure you use correct mathematical terminology
- For example: Stretch vertically by scale factor ½
- Do not use the word "compress" in your exam

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

we-image

Sketch the graph of

y = 2 f (x)

2-5-3-ib-aa-sl-stretch-graph-a-we-solution

Sketch the graph of

y = f (2 x)

2-5-3-ib-aa-sl-stretch-graph-b-we-solution

Stretches of Graphs (DP IB Maths: AI HL)