The graph of each function is a sinusoid. a. Plot the graph of the given function. b. From the g

The graph of each function is a sinusoid. a. Plot the graph...

Key Concepts

Function Transformations

Function transformations involve operations such as scaling, shifting, and reflecting the basic sine or cosine function to obtain variations that model different phenomena. For instance, multiplying by a constant changes the amplitude, while adding a constant shifts the function vertically. Recognizing these transformations is fundamental to converting algebraic expressions into their graphical representations and vice versa.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their oscillatory behavior over one or more periods, highlighting key features like maximum and minimum values, zeros, and symmetry. This process helps in visually understanding the function's amplitude, period, and phase shift. A correct graph enables one to infer properties of the function and to determine corresponding equations from visual patterns.

Algebraic Verification of Sinusoidal Form

Algebraic verification entails showing that a function can be rewritten into the standard sinusoidal form, either in terms of sine or cosine. This involves using trigonometric identities and algebraic manipulations to reveal the function's amplitude, period, phase, and vertical shift, confirming that the function exhibits the defining characteristics of a sinusoid.

Sinusoidal Functions

Sinusoidal functions are periodic functions that take the form of sine or cosine waves. They are characterized by properties such as amplitude, period, phase shift, and vertical shift, and they model many natural phenomena including sound waves, light waves, and tides. Understanding these functions involves recognizing their regular, repeating structure and their smooth, oscillatory behavior.

Trigonometric Identities

Trigonometric identities, such as the power-reducing identities, are key tools in transforming one trigonometric expression into another equivalent form. For example, using the identity sin²x = (1 - cos(2x))/2 allows one to rewrite a squared sine function into a form that directly shows its sinusoidal behavior. This algebraic manipulation is essential in verifying that a given function is indeed sinusoidal.

Transcript

00:01 In this question, we will be talking about half -argument properties.

00:07 This question gives us that the function 10 times sine squared x is a sine -use soil function of x.

00:14 We're asked to graph this function, which i've done here using desmos, the online software, but you should also do it on paper.

00:23 It's very instructive and helpful.

00:27 And then we're also asked to using the graph, find the equation, of this sinusoid and then verify that it is indeed sinusoidal using algebra.

00:41 Okay, let's get started.

00:45 So i have the function 10 sine squared here, 10 sine squared of x.

00:50 I notice that it ranges from y equals 0 to y equals 10.

00:58 In desmos, if you click these points, you can find their exact coordinates.

01:04 And so i see it goes from 0 to 10.

01:10 And that means that the axis of is the halfway point between those two.

01:15 Y equals 5.

01:18 And so that also tells me that the amplitude is 5, because 10 goes 5 higher than 5, and 0 is 5 lower.

01:29 Another thing i want to know about the sinusoid is its period.

01:32 I want to know how high it is, where the height is, how wide it is, and how shifted it is, where the y intercept is pretty much.

01:43 So since i'm given in the question that the equation is sinusoidal, i can start with either cosx or sine x and then transform them to get this red function.

02:01 Notice that at x equals 0, my red target function has a valley, has a minimum, and the close function has maximum.

02:15 A sign doesn't have either.

02:18 And this tells me that if i use sign, i will have to shift left or right.

02:23 But if i use coes, all i have to do is flip it to get the same property with the coes function as with the red target function.

02:36 That there is a peak at x equal 0.

02:39 And so i'll start with this.

02:42 Now, cos has an amplitude of 1, just by itself, like here, but i want an amplitude of 5, so i will stretch it by 5.

02:54 It also has a period, close of x, that it has a period of 2 pi, but i want it to have a period of 1 pi...

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