Transforming Trig Functions: Amplitude, Frequency, Period, Phase Shifts

Hi everyone, and welcome to MathSux! In this post, we are going to break down transforming trig functions by identifying their amplitude, frequency, period, horizontal phase shift, and vertical phase shifts. Fear not! Because we will break down what each of these terms (amplitude, frequency, period, horizontal phase shift and vertical phase shifts) mean and how to find them when looking at a trigonometric function and then apply each of these changes step by step to our graph. In this particular post, we will be transforming and focusing on a cosine function, but keep in mind that the same rules apply for transforming sine functions as well (example shown below in practice).

And if you’re ready for more, check out the video and the practice problems below, happy calculating! 🙂

*For a review on how to derive the basic Trig functions (y=sinx, y=cosx, and y=tanx), click here.

What are the Different Parts of a Trig Function?

When transforming trig functions, there are several things to look out for, let’s take a look at what each part of a trig function represents below:

Trig functions

Amplitude: The distance (or absolute value) between the x axis and the highest point on the graph.

Frequency: This is the number of cycles that happen between 0 and 2π. (Α “cycle” in this case is the number of “s” cycles for the sine function).

Period: The x-value/length of one cycle. (Α “cycle” in this case is the number of “s” cycles for the sine function). This is found by looking at the graph and seeing where the first cycle ends, or, by using the formula:       

Horizontal Shift: When a trigonometric function is moved either left or right along the x axis.

Vertical Shift: When a trigonometric function is moved either up or down along the y axis.

Let’s try an Example, by graphing the following trig function step by step by identifying the amplitude, frequency, period, vertical shift, and horizontal phase shift.

Step 1: First let’s label and identify all the different parts of our trig function and what each part represents.

Trig functions

Step 2: Now let’s transform our trig graph one step at a time.  First, let’s start graphing y=cos(x) without any transformations, the basic graph.

Trig functions
Graph y=cos(x)

Step 3: Next, let’s add our amplitude of 2, otherwise known as the height, or distance to the x-axis.  To do this our highest and lowest points on the y-axis will now be moved to 2 and -2 respectively. 

Trig functions
Re-draw y=cos(x), with an amplitude of 2 to get y=2cos(x)

Step 4: Next, we can apply our horizontal shift to the left by (π/2) or 90º.  To do this, we need to look at where negative (π/2) is on our graph at (-π/2) and move our entire graph over to start at this new point, “shifting” over each coordinate point by (π/2) along the x axis.

Trig functions
Shift our graph y=2cos(x) over by 90º to the left to get y=2cos(x+(π/2))

Step 5: For our last transformation, we have a vertical phase shift up 1 unit.  All this means is that we are going to shift our entire graph up by 1 unit along the y axis.

Trig function graph
phase shift
vertical shift
Shift our entire graph y=2cos(x+(π/2)) up one unit along the y-axis to get y=2cos(x+(π/2))+1

Think you are ready to try graphing trig functions and identifying the amplitude, frequency, period, vertical phase shift, and horizontal phase shift? Check out the practice questions and answers below!

Practice Questions:

When you’re ready check out the function transformations solutions below:

Solutions:

Still, got questions? No problem! Don’t hesitate to comment with any questions or check out the video above for an in depth explanation. Happy calculating! 🙂

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Related Posts:

If you’re wondering how we graphed y=cos(x) and y=sin(x), please check out these related posts on trig graphs and the unit circle below:

Graphing y=cos(x), y=sin(x), y=tan(x)

Deriving the Unit Circle

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