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

Graphical Analysis of Trigonometric Functions

Graphical analysis of trigonometric functions involves studying the key features of the graph, such as amplitude (the height of the peaks), period (the distance required for the function to complete one cycle), phase shift (the horizontal displacement), and vertical shift. By analyzing these features, one can deduce the corresponding equation that represents the graph. This analysis is essential in applications where the function is derived from observed data, and its parameters need to be determined visually.

Trigonometric Identities

Trigonometric identities are fundamental tools used to simplify and transform expressions involving trigonometric functions. Identities such as the sum-to-product formulas and the angle addition formulas allow for the rewriting of sums of sine and cosine terms into a single sinusoidal expression. This is crucial for verifying that a given combination of trigonometric functions retains the sinusoidal shape and for algebraically confirming properties of the function.

Sinusoidal Function

A sinusoidal function is a periodic function that can be expressed as a transformation of the basic sine or cosine function. It typically takes the form y = A sin(B(x - C)) + D or y = A cos(B(x - C)) + D, where A represents the amplitude, B affects the period, C indicates the phase shift, and D is the vertical shift. These functions are used to model repetitive phenomena such as sound waves, light waves, and seasonal patterns.

Amplitude-Phase Transformation

The amplitude-phase transformation is a technique used to rewrite a linear combination of sine and cosine functions into a single sinusoidal function of the form R cos(x - ?) or R sin(x + ?). This method involves finding the amplitude R = ?(a² + b²) and the phase shift ?, enabling a clearer understanding of the function's behavior in terms of its maximum value and horizontal displacement. It simplifies both the algebraic manipulation and the graphical analysis of sinusoidal functions.

Transcript

00:01 In this question we will be talking about combinations of sinusoids.

00:06 This question asks us to graph this function, cosine of x plus sine of x, find the equation of this sinusoid using the graph, and then check our work using algebra.

00:23 So we're given that this function is a sinusoidal function, and we want to find its equation.

00:30 I've already graphed it because i have to start the question somewhere.

00:34 I've graphed on this desmos website, and that allows me to locate these points, significant points on the graph, which tell me things about the equation.

00:49 Now, first of all, notice that the equation ranges, or the function, its y coordinates range from 1 .414 to negative 1 .414.

01:01 And by the way, that's the square root of 2.

01:04 Square root of 2 to negative the square root of 2.

01:07 That tells me the amplitude.

01:11 Also, i know that the function has an axis at y equals 0, the x -axis.

01:20 And that tells me that the amplitude, the distance that the function goes above or below the axis, above or below 0, is root 2.

01:28 So the amplitude is root 2, and the location of the graph is centered on the x -axis.

01:39 Now i have information about the vertical size and position of this function.

01:46 I also want information about the horizontal size and position.

01:53 So horizontal size, as i call it, can be measured using the period.

01:59 Here i see that a period ranging from this point to this point is 8 pi over 4, which is 2 pi, just like the original or parent sign function, sinex.

02:19 So now i know almost exactly what i have to do to this sine function to get it to become this sinex plus coax function.

02:29 But i have to, how i have to transform it, i mean.

02:32 I don't have to change the period because it is already the same, 2 pi.

02:38 I just have to do a shift because this red function starts neither at the middle nor at its peak...

Please give Ace some feedback