Stretches of Graphs (DP IB Applications & Interpretation (AI)): Revision Note

Stretches of Graphs

What are 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

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

• A horizontal stretch of the graph by a scale factor q centred about the y-axis is represented by 

  • 

• The x-coordinates change

  • They are divided by q

• The y-coordinates stay the same

• The coordinates become 

• Horizontal asymptotes stay the same

• Vertical asymptotes change

  • becomes 

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

• A vertical stretch of the graph  by a scale factor p centred about the x-axis is represented by

  • 

  • This is often rearranged to 

• The x-coordinates stay the same

• The y-coordinates change

  • They are multiplied by p

• The coordinates become 

• Horizontal asymptotes change

  • becomes 

• Vertical asymptotes stay the same