Transformations of Trigonometric Graphs

What You’ll Learn

In this lesson you’ll learn how to transform the graphs of sine and cosine functions by changing amplitude, period, phase shift, and vertical shift.

The Concept

The general form for transformed sine and cosine functions is:

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

Each letter controls a different transformation:

• a = amplitude (vertical stretch or compression). The wave reaches |a| above and below the midline.
• b = affects the period. Period = $\frac{2\pi}{b}$. Larger b means the wave repeats faster.
• c = phase shift (horizontal shift). Positive c shifts right, negative shifts left.
• d = vertical shift. Moves the entire wave up or down. The midline becomes y = d.

Quick reference:

Parameter	What it does	Formula
a	Height of the wave	Amplitude = |a|
b	Speed of repetition	Period = 2π / b
c	Horizontal slide	Shift right by c
d	Vertical slide	Midline at y = d

Worked Example

Graph y = 3 sin(2(x - π/4)) + 1

Let’s identify each parameter:

• a = 3, so amplitude = 3
• b = 2, so period = 2π / 2 = π
• c = π/4, so phase shift = π/4 to the right
• d = 1, so midline at y = 1

The faint blue curve is the standard y = sin(x) for comparison. The orange curve is the transformed version. Notice how it’s taller (amplitude 3 vs 1), repeats faster (period π vs 2π), shifted right by π/4, and centered on y = 1 instead of y = 0.

The wave oscillates between y = 1 - 3 = -2 and y = 1 + 3 = 4.

Real-World Application

Transformed trig graphs model many real situations:

• Sound waves with different volumes (amplitude) and pitches (period)
• Seasonal temperature variations (vertical shift for average temp, amplitude for range)
• Tide heights throughout the day (period and phase shift)
• Alternating current in electricity (amplitude and frequency)
• Daylight hours throughout the year

Example: the number of daylight hours in a year can be modeled as a transformed sine function. In a northern city, it might look something like y = 6 sin(2π/365 (x - 80)) + 12, where the amplitude is about 6 hours, the period is 365 days, the phase shift accounts for the spring equinox, and the vertical shift of 12 represents the average.

You’ve Got This

Transformations follow the same rules as other functions: a affects height, b affects how fast it repeats, c shifts horizontally, and d shifts vertically. Start by identifying each parameter, then sketch the basic wave and apply the changes one at a time.

Quiz

In $y = 4\sin(2x) + 1$, the amplitude is:

• A.$2$
• B.$4$
• C.$1$
• D.$\pi$

In $y = \sin\left(3\left(x - \frac{\pi}{6}\right)\right) - 2$, the phase shift is:

• A.$2$ units down
• B.$\frac{\pi}{6}$ to the left
• C.$3$ to the right
• D.$\frac{\pi}{6}$ to the right

The period of $y = \cos(4x)$ is:

• A.$\frac{\pi}{2}$
• B.$2\pi$
• C.$4\pi$
• D.$\pi$

A vertical shift of $+3$ means the midline is at:

• A.$y = 0$
• B.$y = -3$
• C.$y = 3$
• D.$y = 6$

In $y = 2\sin(x) + 5$, the wave oscillates between:

• A.$-2$ and $2$
• B.$3$ and $7$
• C.$5$ and $7$
• D.$3$ and $5$