Family Overview

In Plain Language

Topic 3.6 is about reading and writing transformations of sine and cosine:

$$y=a\sin\big(b(\theta+c)\big)+d \quad\text{or}\quad y=a\cos\big(b(\theta+c)\big)+d$$

• Form: $a\neq 0$; sine and cosine transforms work the same way because cosine is a phase shift of sine (3.6.A.1).
• $+d$: vertical shift of the graph and midline by $d$ (3.6.A.2).
• $(\theta+c)$: phase shift of $-c$ (horizontal translation) (3.6.A.3).
• $a$: vertical dilation; amplitude becomes $|a|$ (3.6.A.4).
• $b$: horizontal dilation; period becomes $2\pi/|b|$ (3.6.A.5).
• Package: amplitude $|a|$, period $2\pi/|b|$, midline shift $d$, phase shift $-c$ (3.6.A.6).

Why It Matters

This is the analytical toolkit for graphing and matching models before contextual modeling in 3.7. AP items ask students to identify $a,b,c,d$ from an equation or graph.

How Families Can Help

• Color-code: $a$ stretch, $b$ period, $c$ phase, $d$ midline.
• Practice: "phase shift is $-c$" when the form is $\sin(b(\theta+c))$.
• Check: period $=2\pi/|b|$, not $2\pi b$.

What Not to Do

• Do not report phase shift as $+c$ when the CED form uses $(\theta+c)$ (shift is $-c$).
• Do not forget absolute values for amplitude and in the period formula.
• Do not mix forms like $\sin(b\theta+c)$ without rewriting to $\sin\big(b(\theta+c/b)\big)$ carefully.