Sketch the graph of the function. (Include two full periods.) y=2 sec3 x

Name: Problem 24, Chapter 4: Precalculus
Uploaded: 2021-02-24T19:16:08-08:00
Duration: 6 min 1 s
Channel: Emily Keffer
Description: Sketch the graph of the function. (Include two full periods.) y=2 sec3 x

Sketch the graph of the function. (Include two full...

Key Concepts

Vertical Asymptotes

Vertical asymptotes in the graph of a trigonometric function occur at the values of x where the function is undefined. For the secant function, these asymptotes occur at the points where the cosine function is equal to zero, leading to discontinuities that are an essential feature in graphing the function accurately.

Secant Function

The secant function is defined as the reciprocal of the cosine function. This reciprocal relationship means that wherever the cosine function is zero, the secant function has vertical asymptotes. The graph of the secant function is characterized by its U-shaped branches that extend vertically away from these asymptotes.

Trigonometric Transformations

Trigonometric functions can be transformed by applying vertical and horizontal scaling factors. In the case of a function like y = A sec(Bx), the entire graph of the secant function is vertically stretched by the factor A and the period is adjusted by the factor B within the argument of the function.

Period of Trigonometric Functions

The period of a standard trigonometric function is the interval over which the function repeats its pattern. For functions such as y = sec x, the period is 2?. When the function is modified to y = sec(Bx), the period becomes 2?/|B|, effectively compressing or stretching the graph horizontally depending on the value of B.

Transcript

00:01 Let's take a look at the function, y equals 2 times secant of 3x and attempt to graph this.

00:08 Okay, when i graph secant graphs, i like to think about what is the reciprocal of that graph? and maybe i start with that.

00:16 Okay, so the reciprocal of secant, any secant function, is cosine.

00:26 Okay, so instead of graphing y equals 2 times secant of 3x, i'm going to start with y equals two times cosine of 3x.

00:38 And i'll look at those points from cosine, and i'm going to use those points to help me graph secant.

00:45 Okay, so let's start with a vertical shift.

00:49 Is there anything being added or subtracted to our graph that would move it up or down? in this case, there's not.

00:56 So we have no vertical shift, and that means our midline is going to stay at the x -axis.

01:05 Then i'm going to look at the amplitude, and amplitude is going to tell me what is the half the height of my graph? well, half the height of a typical cosine graph is one.

01:19 If i stretch that by two, one times two is two.

01:24 So my amplitude would be two, which means from the midline, i need to count up two to get to any of my maximum cosine points.

01:36 And then from the midline i need to count down to to get to any of my minimum cosine points.

01:45 So now i've kind of got a boundary going on.

01:50 And then i want to look at the period.

01:53 For a cosine, our period is found by doing 2 pi over b, where b is that number in front of x, that coefficient.

02:03 So we get 2 pi over 3, which i can't reduce that anymore, so i'll just leave it at 2 pi over 3.

02:12 That means it takes 2 pi over 3 space to complete one full rotation of cosine or to meet all possible y values that cosine could cross through.

02:23 Okay, so we know that a cosine graph, unless reflected, which this is not, will always start at a maximum.

02:33 That means at zero, our cosine graph is going to start at this maximum.

02:39 And since it starts at a maximum, then at 2 pi over 3, the end of that first period, it's also going to end at a maximum.

02:51 And then i can divide my graph into, i like to think of it as four equal sections.

02:56 Okay, so halfway between 0 and 2 pi over 3 is 1 pi over 3.

03:02 Halfway between 0 and pi over 3 is pi over 6, and then pi over 2 between pi over 3 and 2 pi over 3.

03:10 So i've got four equal sections here, and what's nice about those sections is that each one is going to shift between critical points on that cosine graph...

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