📈 Ride the Waves! Mastering Trigonometric Graphs

Key Features and Transformations

• **Amplitude ($|a|$)** – stretches ($|a| > 1$) or compresses ($0 < |a| < 1$) the graph vertically and flips it if $a<0$.
• **Period** – given by $\dfrac{2\pi}{|b|}$ for $\sin$ and $\cos$, and $\dfrac{\pi}{|b|}$ for $\tan$. A larger $|b|$ squeezes the graph horizontally.
• **Vertical shift ($c$)** – moves the graph up ($c>0$) or down ($c<0$).
• **Phase shift** – although not in $a\sin(bx)+c$ directly, replacing $x$ with $(x-d)$ slides the graph $d$ units right.
• **Asymptotes for $y=a\tan(bx)$** – occur at $bx = \dfrac{\pi}{2}+k\pi$, so they move as $b$ changes.

Worked Example: Sketching $y = 2\sin 3x - 1$