Sinusoidal Functions & Graphs (DP IB Applications & Interpretation (AI): SL): Revision Note

Sinusoidal Functions & Graphs

What are the key features of sinusoidal graphs?

• A sinusoidal graph is of the form

  • $y=a\sin \left(bx\right)+d$

  • $y=a\cos \left(bx\right)+d$

• The y-intercept is at the point where $x=0$

  • $(0,d)$ for $y=a\sin \left(bx\right)+d$

  • $(0,a+d)$ for $y=a\cos \left(bx\right)+d$

• The period of the graph is the length of the interval of a full cycle

  • This is $\frac{360°}{b}$

• The maximum value is $y=a+d$

  • You can find this by letting $\sin \left(bx\right)=1$ or $\cos \left(bx\right)=1$

• The minimum value is $y=-a+d$

  • You can find this by letting $\sin \left(bx\right)=-1$ or $\cos \left(bx\right)=-1$

• The principal axis is the horizontal line halfway between the maximum and minimum values

  • This is $y=d$

• The amplitude is the vertical distance from the principal axis to the maximum value

  • This is the value of $a$