Sketching the Graph of a Trigonometric Function

Transforming Trigonometric Functions

y = f(x) becomes y = af(b(x - h)) + k
- plug in "b(x - h)" in for x
- multiply "a" at the front, and add "k" to the end
y = sin(x) becomes y = a*sin(b(x - h)) + k
y = cos(x) becomes y = a*cos(b(x - h)) + k
y = tan(x) becomes y = a*tan(b(x - h)) + k
"a" represents the amplitude
- in trig functions, amplitude is the main measure of vertical distance
- in y = sin(x) or y = cos(x), amplitude = 1, so in y = a*sin(x) or y = a*cos(x), amplitude = 1 * a = a
- if "a" is negative, the function has been reflected on the centre line
"b" represent the period
- in trig functions, period is the main measure of horizontal distance
- in y = sin(x) or y = cos(x), period = 2pi, so in y = sin(b*x) or y = cos(b*x), period = 2pi * 1/b = 2pi/b
"h" represents the phase shift
- in trig functions, horizontal shift is called phase shift (different name, same idea)
- if h > 0, shift h units right
- if h < 0, shift h units left
"k" represents the centre line
- in y = sin(x) or y = cos(x), the centre line is y = 0, so in y = sin(x) + k or y = cos(x) + k, the centre line is y = k

first, find the new centre line (to get an idea of where the function will be)
- the new centre line is y = k
next, find the minimum and maximum
- centre line + amplitude = maximum
- centre line - amplitude = minimum
next, find the starting point (the first point you will plot)
- sine functions start at the centre line
- cosine functions start at the maximum (or the minimum if reflected vertically, "a" is negative)
- move the point left or right based on the phase shift
next, find the period (the horizontal distance until the function repeats)
- copy the starting point every period
plot and connect points

find the values of "a", "b", "h", and "k" in y = a*sin(b(x - h)) + k or y = a*cos(b(x - h)) + k
a = amplitude (distance from centre line to max/min)
b = 2pi/period (plug in the period of the function)
h = phase shift (sin starts in middle, cos starts at top)
k = centre line (y = k)