Section+1.11+Sinusoidal+Functions+and+Their+Graphs

**Section 1.11: Sinusoidal Functions and Their Graphs**
-Ms. Tyson handed back quizzes and announced that there will be another next week. -We went over another example of the problem. -The class had a discussion about our grades online and the grading in the class in general. -After consulting Mr. Sutton, the plurality of the class (12 out of 29 students) voted to use the discussion tab for comments. -We went over a few problems in class to help gain a better understanding of sinusoidal functions and their graphs.
 * In Class:**

**Quiz problem example:**
math $

cos(x-1)=\tfrac{-1}{3}

x-1=\cos^{-1}(\frac{-1}{3})

x-1=1.91+2\pi(k); k\epsilon\mathbb{Z}

2\pi -1.91+2\pi (k); k\epsilon\mathbb{Z}

x=2.91+2\pi (k); k\epsilon\mathbb{Z}

x=5.37+2\pi (k); k\epsilon\mathbb{Z}

$

math Keep in mind that the moment when you take the inverse cosine, there are two solutions.


 * Classroom Examples:**

Proper form: Asin(Bx+C)+D A=amplitude B=number of cycles from 0 to 2 π C=phase shift D=vertical displacement (and where the axis of oscillation lies)

1. Graph various functions and compare their periods. a) Graph of f(x)=sin2x, period= 2/10(2 π)

b) Graph of f(x)=sin3x, period= 3/10(2 π) c) Graph of f(x)=sin4x, period=4/10(2 π)

d) Graph of f(x)=sin10x, period=1/10(2 π)

Notice how the period gradually ges smaller in this problem. Furthermore, the graphs show a distinct relationship to the period of y=sin(x)

2. Graph of f(x)=5sin(3x)+2

Notice how the period, horizontal stretch, and horizontal shrinkage are affected.
 * Graphs should include at least nine points

**Important Period Information:**
Period of y=sin(x)=2π Period of y=cos(x)=2π Period of y=tan(x)=π

Given an equation in the form of y=Asin(Bx+C)+D, where A=amplitude, B=number of cycles the graph completes in an interval from 0 to 2 π, C=phase shift, and D=vertical displacement, find the period by using 2π/B. When B increases, the period of the function decreases. Furthermore, only variable B directly affects the period of the function.

Given a graph with enough information, you can create an equation for it by finding the period of the graph. To find a branch of all solutions of a function, you must add the k(period), where k is any integer. Given a point on a sinusoidal function, you can find a point with the same y coordinate on the graph by adding the period to the x coordinate.
 * Application and Uses of Period: **