Combinations of Transformations (SQA National 5 Maths): Revision Note

Combinations of transformations

• Exam questions will usually include multiple transformations of a trigonometric graph

  • So you need to know how to combine the various single transformations

asinbx and acosbx

• Transformations of this form combine a change of amplitude (vertical stretch) and a multiple angle (horizontal stretch)

  • The amplitude and the y-coordinates of minimum and maximum turning points are the same as for $a\sin x$ and $a\cos x$

  • The period and the x-coordinates of roots and minimum and maximum turning points are the same as for $\sin bx$ and $\cos bx$

• For example, the graph of $a\sin bx$ looks like this:

asin(x+d) and acos(x+d)

• Transformations of this form combine a change of amplitude (vertical stretch) and a phase angle (horizontal translation)

  • The period is 360°

  • The amplitude and the y-coordinates of minimum and maximum turning points are the same as for $a\sin x$ and $a\cos x$

  • The x-coordinates of roots and minimum and maximum turning points are the same as for $\sin \left(x+d\right)$ and $\cos \left(x+d\right)$

• For example, when $\mathit{d}$ is positive the graph of $a\sin \left(x+d\right)$ looks like this

  • In this case, $0<d<90$

  • Remember, positive $d$ causes a shift to the left (while negative $d$ would cause a shift to the right)

sin(x+d)+c and sin(x+d)+c

• Transformations of this form combine a vertical translation and a phase angle (horizontal translation)

  • The period is 360°

  • The amplitude is 1

  • The y-coordinates of minimum and maximum turning points and of the 'middle line' are the same as for $\sin x+c$ and $\cos x+c$

  • The x-coordinates of roots and minimum and maximum turning points are the same as for $\sin \left(x+d\right)$ and $\cos \left(x+d\right)$

sin(bx)+c and cos(bx)+c

• Transformations of this form combine a vertical translation and a multiple angle (horizontal stretch)

  • The amplitude is 1

  • The y-coordinates of minimum and maximum turning points and of the 'middle line' are the same as for $\sin x+c$ and $\cos x+c$

  • The period and the x-coordinates of roots and minimum and maximum turning points are the same as for $\sin bx$ and $\cos bx$

asinx+c and acosx+c

• Transformations of this form combine a change of amplitude (vertical stretch) and a vertical translation

  • Both transformations are vertical

  • The change of amplitude happens before the vertical translation

• For example, for $2\sin x-4$

  • There is a stretch by a factor of 2, then a translation 4 units down

    • The amplitude is 2 (vertical translation doesn't affect amplitude)

    • The period is 360° (vertical changes don't affect the period)

  • To find coordinates of important points, multiply the y-coordinate by 2, then add 4 (the x-coordinates don't change)

    • The graph 'starts' at $\left(0,2\times 0-4\right)=\left(0,-4\right)$

    • The first maximum turning point occurs at $\left(90,2\times 1-4\right)=\left(90,-2\right)$

sin(bx+d) and cos(bx+d)

• Transformations of this form combine a multiple angle (horizontal stretch) and a phase angle (horizontal translation)

  • Both transformations are horizontal

  • The horizontal translation (phase angle) happens before the horizontal stretch (multiple angle)

• For example, for $\cos \left(2x+30\right)$

  • There is a translation 30° left, then a stretch by a factor of $\frac{1}{2}$

    • The amplitude is 1 (horizontal changes don't affect amplitude)

    • The period is $\frac{360}{2}$=180° (horizontal translation doesn't affect period)

  • To find coordinates of important points, subtract 30 from the x-coordinate by 2, then divide by 2 (the y-coordinates don't change)

    • The graph 'starts' at $\left(\frac{0-30}{2},1\right)=\left(-15,1\right)$

    • The first minimum turning point occurs at $\left(\frac{180-30}{2},-1\right)=\left(75,-1\right)$

Finding equations or coordinates of transformed graphs

How do I find equations or coordinates of points from transformed trigonometric graphs?

• On the exam, questions on this topic are usually of one of two forms

  • You are given a graph with definite coordinates, and asked to find the corresponding equation in a certain form

  • You are given the trigonometric equation and asked to find the coordinates for a particular point on the corresponding graph

Finding equations

• To find an equation, the key is to

  • compare features of the graph (for example the period, the amplitude, or the coordinates of a turning point)

  • with what those features should be in terms of the variables in the equation

• For example, if you know the equation is in the form $y=a\cos bx$, and you can see that the graph has a maximum turning point at (0, 7) and completes two cycles between 0 and 360°

  • The graph of $\cos x$ has been vertically stretched by a factor of $a$ and horizontally stretched by a factor of $\frac{1}{b}$ to get the graph of $a\cos bx$

  • The amplitude of $a\cos bx$ is $a$, and it should therefore have a maximum point at $\left(0,a\right)$

    • So $a=7$

  • The period of $a\cos bx$ is ${\left(\frac{360}{b}\right)}^{°}$ and the period of the graph is 180° (2 cycles in 360°)

    • So $180=\frac{360}{b}\Rightarrow b=\frac{360}{180}=2$

  • The equation is $y=7\cos 2x$

Finding coordinates

• To find coordinates

  • determine what the coordinates of the point on the graph should be

  • based on the numbers used in the equation

• For example, if you know that the equation is $y=3\sin \left(x-60\right)$, and you are asked to find the coordinates of the first maximum point

  • The graph of $\sin x$ has been vertically stretched by a factor of 3 and horizontally translated 60° to the right to get the graph of $3\sin \left(x-60\right)$

  • So instead of being at (90°, 1), the first maximum point should be at

    • $\left(90-\left(-60\right),1\times 3\right)=\left(150°,3\right)$