Periods & Amplitudes (Cambridge (CIE) IGCSE International Maths: Extended): Revision Note

Periods & Amplitudes

What are oscillating graphs?

• The sine and cosine graphs are oscillating graphs

  • Both graphs oscillate between a maximum and a minimum point

  • A cycle is completed when the graph returns an initial starting point

  • The cycle is repeated indefinitely in both the positive and negative x directions

What are the features of oscillating graphs?

• An oscillating graph has the following key features:

  • The amplitude is half the vertical distance from the maximum point to the minimum point

    • This is the distance from the horizontal line of symmetry (often the x-axis) to the minimum or maximum point

  • E.g. For the graph $y=\sin x$ the amplitude is 1

• The period is the distance along the x-axis to complete a full oscillation

  • Maximum point to maximum point

  • Minimum point to minimum point

  • x-intercept to next but one x-intercept

  • E.g. For the graph $y=\cos x$ the period is 360º

How do I draw an oscillating graph from its equation?

• An oscillating function has equation

  • $\mathrm{f}\left(x\right)=a\sin \left(bx\right)$

  • $\mathrm{f}\left(x\right)=a\cos \left(bx\right)$

• The $a$ represents the amplitude of the function

  • The bigger the value of $a$ the bigger the range of values of the function

  • E.g. For the graph $y=5\cos x$

    • The amplitude is 5 units

    • The distance from the x-axis (horizontal line of symmetry) to the maximum/minimum point would be 5 units

  • E.g. For the graph $y=2+\frac{1}{5}\cos x$

    • The amplitude is $\frac{1}{5}$ of a unit

    • The vertical distance from the line $y=2$ to the maximum/minimum point would be $\frac{1}{5}$ of a unit

• The $b$ determines the period of the function

  • The bigger the value of $b$ the quicker the function repeats a cycle

  • The period is $\frac{360°}{b}$

    • A larger value of $b$ produces a shorter period

  • E.g. For the graph $y=\sin \left(2x\right)$

    • The period is $\frac{360}{2}=180°$

    • The graph has been 'squashed' by a factor of 2 horizontally

    • Two complete cycles will occur in an interval of $360°$

  • E.g. For the graph $y=\sin \left(\frac{1}{2}x\right)$

    • The period is $\frac{360}{{\displaystyle \frac{1}{2}}}=360\times \frac{2}{1}=720°$

    • The graph has been 'stretched'

    • Half a cycle will occur in an interval of $360°$