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 refer to changes applied to the basic trigonometric functions such as scaling, shifting, or reflecting the graph. When a trigonometric function is modified by operations like squaring or by altering its argument, these transformations affect the amplitude, period, phase shift, and vertical position of the graph. Understanding these transformations is key to accurately graphing and interpreting trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. They are essential tools for simplifying expressions and for rewriting functions in different but equivalent forms. In the context of verifying that a function is sinusoidal, identities such as the double angle formulas or power-reduction formulas can be used to express a function in the standard sinusoidal form.

Algebraic Verification

Algebraic verification involves using algebraic manipulations and trigonometric identities to confirm that a function conforms to a particular form or possesses certain properties. In the context of sinusoidal functions, it means demonstrating through algebraic steps that an expression can be equivalently written in the standard form of a sinusoid. This process reinforces the understanding of the relationship between different trigonometric representations and their graphical behavior.

Sinusoidal Functions

Sinusoidal functions are functions that have the same shape as the sine or cosine curves. They are characterized by parameters such as amplitude, period, phase shift, and vertical shift, which control the height, width, and position of the wave. These functions are fundamental in modeling periodic phenomena in various fields such as physics, engineering, and signal processing.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting the wave-like behavior of sine, cosine, or their transformed versions on a coordinate system. This process includes identifying key features like maximum and minimum points, intercepts, and the period of the function. Visualizing these characteristics helps in understanding the periodic nature and the effects of any transformations applied to the base functions.

Transcript

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

00:08 Here we're given that the graph of co -square of 3x, this graph here, is sinusoidal.

00:18 So i've already plotted it using this plotting software, but you can also plot it by hand.

00:26 And given the graph, we want to find the equation of the sinusoid and verify using algebra, that is indeed a sinusoid.

00:39 So zooming in here, we can see that the function ranges from the value, the y value of 1 to the y value 0.

00:51 So the function is always between 0 and 1.

00:56 The range is from 0 to 1.

01:00 And that tells us that the halfway point is 1 half.

01:07 So that tells us about two properties of the function, the axis of symmetry of the sinusoid, and the amplitude, which is the distance it goes above and below this axis.

01:26 We also need two more characteristics about the sinusoid to determine its exact equation.

01:35 In a sense, its horizontal width and its horizontal position.

01:43 The two properties i already talked about, the axis of symmetry and the amplitude, tell us about the vertical position and the vertical width.

01:55 So we notice that between two crests of this periodic function, the distance is pi over 3.

02:05 So from minus pi over 3 is 0 or from 0 to pi over 3.

02:09 So that tells us that the period is pi over 3 rather than 2 pi, like the original cos of x function.

02:21 And that means to get the period to be this, to use, to transform this cosine of x function, have the same period as this red graph, we want to compress it by a factor of 6.

02:36 So 3 pi over 3 is 6 times smaller than 2 pi, which is the period of the original function i had.

02:45 So now i have the period the same, but i don't have the amplitude the same, nor the axis of symmetry.

02:54 I do have the horizontal position the same because they both have the same y intercept...

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