Trig Functions (Amp, Freq, Phase Shifts): Algebra 2/Trig.

Hi everyone, and welcome to MathSux! In this post we are going to break down how to graph trig functions by identifying its amplitude, frequency, period, and horizontal and vertical phase shifts. Fear not! Because we will breakdown what each of these mean and how to find them, then apply each of these changes step by step on our graph. 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 basic Trig functions (y=sinx, y=cosx, and y=tanx), click here.

What are the Different Parts of a Trig Function?

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, graphing a Trig Function step by step.

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

Trig functions

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

Trig functions

Step 3: Let’s add our amplitude of 2, the 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

Step 4: Next, we do a horizontal phase shift to the left by (π/2).  To do this, we 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” it to (π/2).

Trig functions

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 functions

Practice Questions:


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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