Lesson 4.5

Vertical Shifts

Adding a constant to a sine or cosine function shifts the entire wave up or down. The midline moves, but the shape stays the same.

Introduction

Many physical phenomena don't oscillate symmetrically around zero. Temperature fluctuates around a seasonal average, a Ferris wheel rotates around a hub height above the ground — the vertical shift models this.

Past Knowledge

You know amplitude and period, and can graph sinusoidal functions with midline .

Today's Goal

Graph and identify the midline, maximum, and minimum.

Future Success

Vertical shift is the third parameter in .

Key Concepts

How Vertical Shifts Work

For :

Midline:
Maximum:
Minimum:

Visual Comparison

Gray = sin(x) · Blue = sin(x) + 3 · Red = sin(x) − 2

Finding D from a Graph

The midline is always the average of the maximum and minimum values.

Worked Examples

Basic

Identifying the Midline

Question: Find the midline, maximum, and minimum of .

Step 1: , .

Step 2: Midline: . Max: . Min: .

Final Answer: Midline , max , min .

Intermediate

With Amplitude and Shift

Question: Find the range of .

Step 1: , .

Step 2: Max: . Min: .

Final Answer: Range:

Advanced

Finding A and D from Max/Min

Question: A cosine wave has max and min . Find and .

Step 1: Amplitude:

Step 2: Midline:

Final Answer:

Common Pitfalls

Confusing Vertical Shift with Amplitude

In , the amplitude is (not ). The wave oscillates units above and below the midline . The moves the center, not the height.

Real-Life Applications

Ferris Wheel Height

A Ferris wheel with a radius of meters and a hub height of meters creates a rider-height function like . The shifts the midline up to the hub height, so the rider oscillates between m and m above the ground.