Topic 3.6 Notes – Sinusoidal Function Transformations

What Sinusoidal Transformations Are

A transformed sinusoidal function has one of these forms:

$$f(\theta)=a\sin(b(\theta+c))+d \qquad g(\theta)=a\cos(b(\theta+c))+d$$

Your parent reminder is simple. For both $y=\sin\theta$ and $y=\cos\theta$:

• amplitude $=1$
• period $=2\pi$
• midline $y=0$

The four parameters work together:

• $a$ changes vertical stretch or compression, and if it’s negative, the graph reflects
• $b$ changes horizontal stretch or compression, so it changes the period
• $c$ gives the phase shift, which is the horizontal shift
• $d$ gives the vertical shift, so the midline becomes $y=d$

In the graph, each colored curve changes just one feature of $y=\sin\theta$. That makes it easier to connect the equation to what happens visually.

Basic transformations of $y=\sin\theta$

Sine and cosine follow the same transformation rules because cosine is just a shifted sine curve. Also, formulas here use radians, not degrees.

Reading the Parameters from an Equation

For $a\sin(b(\theta+c))+d$ or $a\cos(b(\theta+c))+d$:

$$\text{amplitude}=|a|,\quad \text{period}=\frac{2\pi}{|b|},\quad \text{phase shift}=-c,\quad \text{midline } y=d$$

And from that you also get:

• maximum $=d+|a|$
• minimum $=d-|a|$
• range $=[d-|a|,\ d+|a|]$
• domain is all real numbers

Reflections and sign effects

A few signs trip people up a lot:

• If $a<0$, the graph is reflected relative to its midline shape. The amplitude is still positive.
• If $b<0$, the period still uses $|b|$.
• A negative $b$ can often be rewritten away, so for period, your eye should go to $|b|$.

Factoring before finding phase shift

If the inside is written as $B\theta+C$, don’t grab $C$ as the shift.

$$B\theta+C=B\left(\theta+\frac{C}{B}\right)$$

So the phase shift is $-\frac{C}{B}$, not $-C$.

Example:

$$-2\cos(4\theta+\pi)+1 = -2\cos\left(4\left(\theta+\frac{\pi}{4}\right)\right)+1$$

Now you can read it correctly:

• amplitude $=2$
• period $=\frac{2\pi}{4}=\frac{\pi}{2}$
• phase shift $=-\frac{\pi}{4}$ or right $\frac{\pi}{4}$
• vertical shift $=1$
• midline $y=1$

Sketching One Cycle from the Equation

One cycle is easiest with five key points.

For example, the graph of $y=2\sin(\theta-\frac{\pi}{4})+1$ starts at the phase shift, then hits the next four key points by moving one quarter-period at a time.

Rewrite in the clearer form $a\sin(b(\theta-h))+d$ or $a\cos(b(\theta-h))+d$, where $h$ is the phase shift. Then find:

• midline $y=d$
• amplitude $|a|$
• period $P=\frac{2\pi}{|b|}$
• quarter-period $\frac{P}{4}$

Place the first point at $\theta=h$, then move by quarter-periods.

Sine pattern

• at $h$, value $d$
• at $h+\frac{P}{4}$, value $d+a$
• at $h+\frac{P}{2}$, value $d$
• at $h+\frac{3P}{4}$, value $d-a$
• at $h+P$, value $d$

Positive sine rises first. Negative sine falls first.

Cosine pattern

• at $h$, value $d+a$
• at $h+\frac{P}{4}$, value $d$
• at $h+\frac{P}{2}$, value $d-a$
• at $h+\frac{3P}{4}$, value $d$
• at $h+P$, value $d+a$

Positive cosine starts at a maximum. Negative cosine starts at a minimum.

Writing a sinusoid from a sketched graph

Work from the graph’s visible features.

$$\text{amplitude}=\frac{M-m}{2} \qquad d=\frac{M+m}{2} \qquad b=\frac{2\pi}{P}$$

where $M$ is the maximum, $m$ is the minimum, and $P$ is the period.

Then choose a form that matches an easy feature:

• maximum $\rightarrow$ positive cosine
• minimum $\rightarrow$ negative cosine
• increasing midline crossing $\rightarrow$ positive sine
• decreasing midline crossing $\rightarrow$ negative sine

Example with maximum $8$, minimum $2$, period $3\pi$, and maximum at $-\frac{\pi}{4}$:

• amplitude $=\frac{8-2}{2}=3$
• midline $d=\frac{8+2}{2}=5$
• $b=\frac{2\pi}{3\pi}=\frac{2}{3}$

A good equation is

$$y=3\cos\left(\frac{2}{3}\left(\theta+\frac{\pi}{4}\right)\right)+5$$

Common Mistakes and Exam Traps

• Amplitude is $|a|$, so it is never negative.
• Period is $\frac{2\pi}{|b|}$, not $\frac{2\pi}{b}$ if $b<0$.
• In $4\theta+\pi$, the phase shift is not just $-\pi$. Factor first.
• Inside and outside shifts go different directions. $+d$ shifts up, but $+c$ inside shifts left.
• The midline $y=d$ is the center of the wave, not the max or min.
• A graph can have more than one correct equation. Equivalent sine and cosine forms are common on FRQs and MCQs.