Today's soundtrack is The Everly Brothers: Bye Bye Love, a compilation of their hits.
This morning, I'm continuing where we left off yesterday in learning about the various transformations that we can apply to trigonometric functions.
So we already know how to phase shift (horizontally translate), change the amplitude (vertically expand or compress), and vertically displace (translate vertically). Now, we'll be learning how to expand or compress a trigonometric function horizontally.
We'll use the variable b to modify the horizontal compression or expansion of our function. It's placed into the equation like this: y=sin(bx). Note that b, which modifies x, is on the same side of the equation as x; thus, its effects on x will be the inverse of what they would first appear. For example, y=sin(2x) will be compressed by a factor of 1/2; y=sin(1/2x) will be expanded by a factor of 2.
Modifications to b: Resulting Transformations
If b > 1, compress horizontally by a factor of 1/b
If 0 < b < 1, expand horizontally by a factor of b
If 0 > b > -1, reflect on the y-axis and expand horizontally by a factor of b
If b = -1, reflect on the y-axis
If b < -1, reflect on the y-axis and compress horizontally by a factor of 1/b
When we apply horizontal expansions and compressions, we say that these change the period of the function. A standard function's period will start at 0 and end at 2π, so we say that its period is 2π. A function with b=2 will be compressed by 1/2, so it will start at 0 and end at π; thus, we will say that its period is π. A function with b=1/2 will be expanded by a factor of 2, so it will start at 0 and end at 4π. Therefore, its period will be 4π.
Without needing to plot a trigonometric function, we can determine its period by using this formula: p=(2Ï€/b), b>0.
Recall that we can also determine the phase shift of a trigonometric function based on its equation. An important note: if we are working with an equation that has both a period change and a phase shift, and b is inside the brackets, we must factor out the value of b so that b=1 before identifying the value of our phase shift.
When we go to graph a trigonometric function, our first step is to identify the following four transformations:
Amplitude
Vertical displacement
Period
Phase shift
That's it for today!