Sinusoidal GraphsJEE Main

Concept

Every sinusoid is a stretched, shifted sine wave: $y = A\sin(Bx + C) + D$. Four knobs control it — amplitude $|A|$, period $2\pi/|B|$, phase shift $-C/B$, and vertical shift $D$ (the midline).

Key formula

Derivation

Start from $\sin x$. Multiplying by $A$ scales the height to amplitude $|A|$. Replacing $x$ by $Bx$ compresses the period to $2\pi/|B|$. Adding $C$ inside shifts the graph left by $C/B$ (so the phase shift is $-C/B$). Adding $D$ outside raises the whole curve.

Cosine is just sine advanced by $90°$: $\cos x = \sin(x + \pi/2)$.

Scenarios to explore

• Sinusoidal Graphs — Amplitude, period and phase of sin & cos.

Real-world applications

• Modelling AC voltage, sound and light waves.
• Simple harmonic motion displacement.
• Tides, seasons and any periodic data.

JEE exam tips

• Max occurs when the inside equals $\pi/2$ (for sine); set $Bx+C = \pi/2$ and solve.
• Range is always $[D-|A|,\;D+|A|]$ regardless of $B$ or $C$.

Common mistakes

• Reading the phase shift as $+C$ instead of $-C/B$.
• Forgetting $B$ changes the period inversely (bigger $B$ = shorter period).
• Treating a negative $A$ as a phase shift — it is a reflection in the midline.

Exam traps to avoid

• Amplitude is $|A|$ (never negative); the sign of $A$ only flips the curve.
• A vertical shift moves the midline — the max/min move with it, the amplitude does not change.