Transforming Trigonometric Graphs

Start with $y=\sin x$. Multiplying its output by 2 doubles every height; multiplying its input by 2 makes it complete a full cycle in half the angle.

$a$ multiplies the output, $b$ multiplies the angle and $c$ is added to the output. Try changing just one at a time.

• Amplitude is $|a|$: the vertical distance is always non-negative. Negative $a$ reflects the wave in the horizontal axis before the vertical shift.
• Midline is $y=c$; the maximum and minimum are $c+|a|$ and $c-|a|$.
• For non-zero $a,b$, period is $360^\circ/|b|$ or $2\pi/|b|$. For example, $b=1/2$ stretches the graph horizontally by 2; it does not compress it.

To see the period rule, make the inside angle increase by a full turn: $bT=360^\circ$ for positive $b$. The required change in $x$ is $T=360^\circ/b$. Negative $b$ also reflects horizontally. If $a=0$ or $b=0$, the graph is constant and has no smallest positive period.

For a horizontal shift, locate where the inside angle is zero:

The base sine starts at inside angle zero, which now occurs at $x=20^\circ$. The wave shifts right 20°, has period 120°, midline 1 and range from −1 to 3. Quarter-period steps of 30° give:

If written as $\sin(3x-60^\circ)$, factor the 3 first. The shift is 20°, not 60°. A plus inside, such as $\sin(x+30^\circ)$, moves left 30°.

Cosine uses the same rules. Tangent uses period $180^\circ/|b|$ or $\pi/|b|$, but has no amplitude. Find its asymptotes by making the inside angle an odd multiple of 90°.