Transformations of Trigonometric Graphs (SQA National 5 Maths): Revision Note

Amplitude, vertical translation, multiple angle & phase angle

What are transformations of trigonometric graphs?

• You should be familiar with the basic trigonometric graphs for sin, cos and tan

• There are four ways of transforming these graphs that could appear in an exam question:

  • Changing the amplitude (also known as a vertical stretch)

  • Vertical translation (i.e., shifting the entire graph up or down)

  • Using a multiple angle (also known as a horizontal stretch)

  • Using a phase angle (also known as a horizontal translation, i.e. shifting the entire graph to the left or right)

What is a change of amplitude of a trigonometric graph?

• A change of amplitude of a trigonometric graph is of the form $a\sin x$ or $a\cos x$

• The graph of $\mathit{a}\mathbf{sin}\mathit{x}$ is similar to the graph of $\sin x$

  • It has the same 'wave' shape

  • It still has a period of 360° (i.e. it repeats every 360°)

  • However its amplitude changes to $a$

    • I.e. instead of oscillating between heights of 1 and -1, it oscillates between heights of a and -a

    • It has been stretched vertically

  • It goes through the origin (0, 0)

    • Then every 90° it cycles through the heights a, 0, -a, 0, ...

• The graph of $\mathit{a}\mathbf{cos}\mathit{x}$ is similar to the graph of $\cos x$

  • It has the same 'wave' shape

  • It still has a period of 360° (i.e. it repeats every 360°)

  • However its amplitude changes to $a$

    • I.e. instead of oscillating between heights of 1 and -1, it oscillates between heights of a and -a

    • It has been stretched vertically

  • It goes through the point (0, a) on the y-axis

    • Then every 90° it cycles through the heights 0, -a, 0, a, ...

What is a vertical translation of a trigonometric graph?

• A vertical translation of a trigonometric graph is of the form $\sin x+c$ or $\cos x+c$

  • Note that the '$+c$' is outside of the trigonometric function

  • I.e. $\sin x+c$ is the same as $\left(\sin x\right)+c$; it is not the same as $\sin \left(x+c\right)$

    • $\sin \left(x+c\right)$ is an example of a phase angle, not a vertical translation

  • $\sin x+c$ can also be written as $c+\sin x$

• The graph of $\mathbf{sin}\mathit{x}\mathbf{+}\mathit{c}$ is similar to the graph of $\sin x$

  • It has the same 'wave' shape

  • It still has a period of 360° (i.e. it repeats every 360°)

  • Its amplitude is still 1

    • I.e. it still oscillates between 1 above and 1 below its 'middle line'

  • However its middle line moves from $y=0$ (the x-axis) to $y=c$

    • It oscillates between heights of c+1 and c-1

    • It has been translated (i.e. shifted) by |c| units vertically

      • If c is positive then it is shifted up

      • If c is negative then it is shifted down

  • It goes through the point (0, c) on the y-axis

    • Then every 90° it cycles through the heights c+1, c, c-1, c, ...

• The graph of $\mathbf{cos}\mathit{x}\mathbf{+}\mathit{c}$ is similar to the graph of $\cos x$

  • It has the same 'wave' shape

  • It still has a period of 360° (i.e. it repeats every 360°)

  • Its amplitude is still 1

    • I.e. it still oscillates between 1 above and 1 below its 'middle line'

  • However its middle line moves from $y=0$ (the x-axis) to $y=c$

    • It oscillates between heights of b+1 and b-1

    • It has been translated (i.e. shifted) by |c| units vertically

      • If c is positive then it is shifted up

      • If c is negative then it is shifted down

  • It goes through the point (0, c+1) on the y-axis

    • Then every 90° it cycles through the heights c, c-1, c, c+1, ...

What is a multiple angle with a trigonometric graph?

• A multiple angle with a trigonometric graph is of the form $\sin bx$ or $\cos bx$

  • These could also be written with brackets as $\sin \left(bx\right)$ or $\cos \left(bx\right)$

  • $x$ is multiplied by $b$ before it is put into sin or cos

• The graph of $\mathbf{sin}\mathit{b}\mathit{x}$ is similar to the graph of $\sin x$

  • It has the same 'wave' shape

  • Its amplitude is still 1 (i.e. it still oscillates between heights of 1 and -1)

  • However its period changes to ${\left(\frac{360}{b}\right)}^{°}$ (i.e. it repeats every $\frac{360}{b}$ degrees)

    • It has been stretched horizontally

      • If b>1 then it is 'squished in' (more complete cycles fit into 360°)

      • If 0<b<1 then its is 'stretched out' (it takes more than 360° to complete one full cycle)

  • It goes through the origin (0, 0)

    • Then every ${\left(\frac{360}{b}\right)}^{°}$ it cycles through the heights 1, 0, -1, 0, ...

• The graph of $\mathbf{cos}\mathit{b}\mathit{x}$ is similar to the graph of $\sin x$

  • It has the same 'wave' shape

  • Its amplitude is still 1 (i.e. it still oscillates between heights of 1 and -1)

  • However its period changes to ${\left(\frac{360}{b}\right)}^{°}$ (i.e. it repeats every $\frac{360}{b}$ degrees)

    • It has been stretched horizontally

      • If b>1 then it is 'squished in' (more complete cycles fit into 360°)

      • If 0<b<1 then its is 'stretched out' (it takes more than 360° to complete one full cycle)

  • It goes through the point (0, 1) on the y-axis

    • Then every ${\left(\frac{360}{b}\right)}^{°}$ it cycles through the heights 0, -1, 0, 1, ...

What is a phase angle with a trigonometric graph?

• A phase angle with a trigonometric graph is of the form $\sin \left(x+d\right)$ or $\cos \left(x+d\right)$

  • $d$ is added to $x$ before it is put into sin or cos

• The graph of $\mathbf{sin}\left(\mathit{x}\mathbf{+}\mathit{d}\right)$ is similar to the graph of $\sin x$

  • It has the same 'wave' shape

  • It still has a period of 360° (i.e. it repeats every 360°)

  • Its amplitude is still 1 ((i.e. it still oscillates between heights of 1 and -1)

  • However its 'starting point' moves from the origin (0, 0) to (-d, 0)

    • Note the negative sign in front of the d

    • It has been translated (i.e. shifted) by |d| units horizontally

      • If d is positive then it is shifted left

      • If d is negative then it is shifted right

  • It 'starts' at the point (-d, 0) on the x-axis

    • Then every 90° it cycles through the heights 1, 0, -1, 0, ...

      • I.e. through the points (-d+90, 1), (-d+180, 0), (-d+270, -1), (-d+360, 0), ...

• The graph of $\mathbf{cos}\left(\mathit{x}\mathbf{+}\mathit{d}\right)$ is similar to the graph of $\cos x$

  • It has the same 'wave' shape

  • It still has a period of 360° (i.e. it repeats every 360°)

  • Its amplitude is still 1 ((i.e. it still oscillates between heights of 1 and -1)

  • However its 'starting point' moves from the origin (0, 0) to (-d, 0)

    • Note the negative sign in front of the d

    • It has been translated (i.e. shifted) by |d| units horizontally

      • If d is positive then it is shifted left

      • If d is negative then it is shifted right

  • It 'starts' at the point (-a, 1) on the x-axis

    • Then every 90° it cycles through the heights 0, -1, 0, 1, ...

      • I.e. through the points (-d+90, 0), (-d+180, -1), (-d+270, 0), (-d+360, 1), ...

Summary of single transformations