Lesson 4.7

Putting It All Together

Now we combine every parameter — amplitude, period, phase shift, and vertical shift — into one comprehensive graphing strategy for .

Introduction

Past Knowledge

You learned A (amplitude), B (period), C (phase shift), D (vertical shift) individually.

Today's Goal

Graph fully transformed sinusoids using a systematic 4-step process.

Future Success

This strategy applies to tangent, secant, and all other trig graphs as well.

Key Concepts

The General Form

Parameter	Controls	Formula
	Amplitude (height)
	Period (speed)
	Phase shift (left/right)	units right
	Vertical shift (up/down)	Midline

The 4-Step Graphing Strategy

Identify A, B, C, D — factor out if needed.
Draw the midline at and mark max/min ().
Compute key -values — start at , add quarter-period four times.
Plot and connect — apply the pattern (zero/max/zero/min/zero for sine, max/zero/min/zero/max for cosine).

— A=3, B=2, C=π/4, D=1

Worked Examples

Basic

Extracting Parameters

Question: Identify A, B, C, D for .

Step 1: Factor out :

Step 2: , , ,

Final Answer: Amplitude 2, period , right , midline , range

Intermediate

Full Key Points

Question: Graph .

Step 1:

Step 2: Period = . Quarter = . Midline: . Range:

Step 3: Start at , add four times:

Step 4: Sine pattern scaled by A=3, shifted up by D=1:

Final Answer: A sinusoid oscillating between and , centered at , with period , starting at .

Advanced

Writing an Equation from a Description

Question: Write a cosine equation: max 10, min 2, period , shifted right .

Step 1: ,

Step 2: ,

Final Answer:

Common Pitfalls

Applying Shifts in the Wrong Order

Always extract by factoring out first, then use as the starting -value. Students who skip factoring compute wrong key points.

Real-Life Applications

Temperature Modeling

A city's average daily temperature might follow , meaning: amplitude 20°F, period 365 days, peak shifted to day 30 (late July), centered at a yearly average of 55°F. Every parameter you learned maps to a real physical quantity.