Transformations of Trigonometric Functions (DP IB Analysis & Approaches (AA): SL): Revision Note

Transformations of trigonometric functions

How do I apply a single transformation to a trig graph?

• You can transform a trig graph using

  • a translation

  • a stretch

  • a reflection

• The table below shows the equations of the trig graphs after each transformation

How do I apply multiple transformations to a trig graph?

• You need to be able to apply multiple transformations to draw graphs written in the form

  • $y=a\sin \left(b\left(x-c\right)\right)+d$

• The order for the vertical transformations are:

  • Reflection in the $x$-axis if $a$ is negative

  • Stretch by scale factor $a$

  • Translation $d$ units

    • Up if positive

    • Down if negative

• The order for the horizontal transformations are:

  • Reflection in the $y$-axis if $b$ is negative

  • Stretch by scale factor $\frac{1}{b}$

  • Translation $c$ units

    • Right if it's $x-c$

    • Left if it's $x+c$

How do transformations affect the trig graph?

• The graph $y=a\sin \left(b\left(x-c\right)\right)+d$ and $y=a\cos \left(b\left(x-c\right)\right)+d$ have the properties:

  • The principal axis is $y=d$

  • The amplitude is $\left|a\right|$

  • The period is $\frac{360°}{\left|b\right|}$

  • The phase shift is $c$

• You can use these properties to sketch a transformed trig graph

• e.g. $y=2\sin \left(3\left(x-45°\right)\right)-1$

  • Draw a sine curve without any axes

  • Identify where the $x$-axis should go

    • Label the principal axis $y=-1$

    • Label the maximum points at $y=-1+2=1$

    • Label the minimum points at $y=-1-2=-3$

  • Find the period $\frac{360°}{3}=120°$

    • You can label the intersections with the principal axis temporarily as 0°, 60°, 120°, etc

  • Identify where the $y$-axis should go

    • Temporally put the $y$-axis going through 0°

    • Translate the graph 45° to the right

    • Add 45° to the intersections with the principal axis

• The graph $y=a\tan \left(b\left(x-c\right)\right)+d$ works similarly

  • There is no amplitude

  • The graph has no minimum or maximum points

  • The period is $\frac{180°}{\left|b\right|}$

  • The graph has asymptotes

    • These are located halfway between the intersections of the graph with the principal axis