Stretches (Edexcel IAL Maths)

Spec point — 1.12

Stretches

What are graph stretches?

• When you alter a function in certain ways, the effects on the graph of the function can be described by geometrical transformations
• With a stretch all the points on the graph are moved towards or away from either the x or the y axis by a constant scale factor

What do I need to know about graph stretches?

• The graph of y = af(x)is a vertical stretch of the graph y = f(x) by a scale factor of a, centred on the x axis

  • The x coordinates of points stay the same; y coordinates are multiplied by a
  • Points on the x axis stay where they are
  • All other points move parallel to the y axis, away from (a > 1) or towards (0 < a < 1) the x axis

• The graph of y = f(ax)is a horizontal stretch of the graph y = f(x) by a scale factor of $\frac{1}{a}$, centred on the y axis

  • The y coordinates of points stay the same; x coordinates are multiplied by $\frac{1}{a}$
  • Points on the y axis stay where they are
  • All other points move parallel to the x axis, away from (0 < a < 1) or towards (a > 1) the y axis

• Any asymptotes of f(x) are also affected by the stretch (stretch them as you would stretch the function of a straight line)

• If an asymptote is one of the coordinate axes, or is parallel to the direction of the stretch, however, it will not be affected

Exam Hint

• When you sketch a stretched graph, be sure to indicate the new coordinates of any points that are marked on the original graph.
• Try to indicate the coordinates of points where the stretched graph intersects the coordinate axes (if you don't have the equation of the original function this may not be possible).
• If the graph has asymptotes, don't forget to sketch the asymptotes of the stretched graph as well.

Worked Example