Transformations of Trigonometric Functions
Learn transformations of trigonometric functions for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-01). Starting from the sine, cosine and tangent curves, you stretch, shift and reflect each graph and see how every change in the equation moves or reshapes the curve.
You will learn to read the amplitude, period, phase shift and vertical shift from an equation such as \(y = a\sin(b(x + c)) + d\), describe the changes made to the basic curve, and state the domain and range — the foundation for modelling anything that repeats in a regular cycle.
Theory
Transformations of trigonometric functions reflect, translate and dilate the graphs of \(\sin\), \(\cos\) and \(\tan\). This Year 12 Mathematics Advanced topic (MAV-12-01) shows how to read the amplitude, period, phase shift, vertical shift and range from \(y=a\sin\!\big(b(x-c)\big)+d\), and how to find the equation of a transformed sine or cosine graph. Angles are in radians.
A transformation of a trigonometric function changes its graph by a reflection, a translation or a dilation, while the graph stays periodic (wave-like). Every transformed sine or cosine can be written as \(y=a\sin\!\big(b(x-c)\big)+d\), and each constant controls one feature.
The amplitude \(|a|\) is the vertical dilation — half the distance from the lowest to the highest point. The number \(b\) is a horizontal dilation that sets the period (the length of one full cycle). The constant \(c\) is the horizontal (phase) shift and \(d\) is the vertical shift that moves the centre line to \(y=d\).
A negative value of \(a\) reflects the graph in the \(x\)-axis. The same ideas apply to \(\cos\) and \(\tan\), except that \(\tan\) has no amplitude and a different period.
For \(y=a\sin\!\big(b(x-c)\big)+d\) with \(b>0\) (and likewise for \(\cos\)):
How to analyse a transformed trig function
- Write it in the form \(y=a\sin\!\big(b(x-c)\big)+d\) — factor the coefficient of \(x\) out of the bracket.
- Read the constants \(a\), \(b\), \(c\) and \(d\).
- Apply the rules: amplitude \(=|a|\), period \(=\dfrac{2\pi}{b}\), phase shift \(=c\), vertical shift \(=d\), and range \(=[\,d-|a|,\,d+|a|\,]\).
| \(\text{amplitude}\) | \(=\) | \(|4|=4\) |
| \(\text{period}\) | \(=\) | \(\dfrac{2\pi}{b}=\dfrac{2\pi}{3}\) |
Amplitude \(4\), period \(\dfrac{2\pi}{3}\).
Amplitude \(|a|=2\), vertical shift \(d=-1\).
| \(\text{max}\) | \(=\) | \(d+|a|=-1+2=1\) |
| \(\text{min}\) | \(=\) | \(d-|a|=-1-2=-3\) |
Range \(-3\le y\le 1\).
Read \(a=-1,\ b=2,\ c=\dfrac{\pi}{3},\ d=1\):
• reflection in the \(x\)-axis (\(a<0\));
• horizontal dilation, factor \(\dfrac{1}{2}\) (\(b=2\));
• translation \(\dfrac{\pi}{3}\) right and \(1\) up.
| \(\text{period}\) | \(=\) | \(\dfrac{2\pi}{2}=\pi\) |
The height is \(3\), so \(|a|=3\); it starts at a maximum, so \(a=3\). One cycle ends at \(x=\pi\):
| \(\pi\) | \(=\) | \(\dfrac{2\pi}{b}\) |
| \(b\) | \(=\) | \(2\) |
So \(y=3\cos 2x\).
Common pitfalls
Frequently asked questions
What is the amplitude of a trigonometric function?
The amplitude is how far the graph rises above or falls below its centre line. For y equals a sin(bx) or a cos(bx) the amplitude is the absolute value of a. For example y equals 4 sin 3x has amplitude 4. The amplitude is always positive; a negative value of a just reflects the graph in the x-axis.
How do you find the period of y = sin(bx)?
Divide 2 pi by b. The period of y equals sin(bx) or y equals cos(bx) is 2 pi over b, so a larger b gives a shorter period (a faster wave). For example y equals 4 sin 3x has period 2 pi over 3.
What does the b value do to a sine or cosine graph?
The b value is a horizontal dilation. It squeezes or stretches the graph sideways and sets the period, which is 2 pi over b. A bigger b makes the wave repeat more often; a smaller b stretches it out.
How do you find the equation of a sine or cosine graph?
Read the amplitude off the graph (half the distance between the highest and lowest points) to get a. Measure the length of one full cycle to get the period, then find b from period equals 2 pi over b. Choose sine or cosine to match the starting point, and add any vertical shift d for the centre line.
What is the period of tan x?
The period of y equals tan x is pi, not 2 pi. For y equals tan(bx) the period is pi over b. Unlike sine and cosine, the tangent function has no amplitude because it increases without bound.
What is the range of y = a sin x + d?
The graph oscillates about the line y equals d, reaching up to d plus the absolute value of a and down to d minus the absolute value of a. So the range is from d minus mod a to d plus mod a. For example y equals 2 cos x minus 1 has range from -3 to 1.