How Do You Find the Amplitude, Period, and Equation of the Principal Axis of a Sinusoidal Model?

For a sinusoidal model of the form $f(x)=a\sin(bx)+d$ or $f(x)=a\cos(bx)+d$, the amplitude is (|a|), the period is $\frac{360^\circ}{|b|}$, and the principal axis is $y=d$. In IB Math AI SL, sinusoidal angles are measured in degrees.

The Reasoning / Mechanism

The coefficient $a$ controls the vertical stretch and any reflection. Amplitude measures distance, so it cannot be negative: a negative $a$ reflects the graph about its principal axis but leaves the amplitude as (|a|).

The coefficient $b$ changes how quickly the cycle repeats. Because sine and cosine complete one standard cycle in $360^\circ$, the period $T$ satisfies $|b|T=360^\circ$, giving . The constant translates the graph vertically, so the midline becomes .

Worked Examples

For $f(x)=-3\cos(2x)+5$, the amplitude is $|-3|=3$, the period is $$, and the principal axis is . Therefore, the model oscillates from to . The negative sign causes a reflection, not a negative amplitude.

Suppose instead that a graph has maximum $11$, minimum $3$, and repeats every $120^\circ$. Then the amplitude is $\frac{11-3}{2}=4$, the principal axis is , and . A suitable model could therefore use , , and , depending on its starting point.

Exam Technique

This is SL 2.5: Modelling functions. In an IB response, state the three features explicitly and write the principal axis as an equation, $y=d$, rather than giving only $d$. Show the substitution before using your GDC and keep it in degree mode.

Common misconception: students sometimes swap amplitude and principal axis, or report a negative amplitude. Check using $\text{maximum}=d+|a|$ and $\text{minimum}=d-|a|$. Give numerical answers exactly or to three significant figures.