Unit 3: Modeling Sinusoidal and Periodic Functions Study Notes

Overview of Unit Three Concepts and Modeling Periodic Phenomena

Session Scope: - This session focuses on modeling sinusoidal functions (sine and cosine) specifically from tables of data. - Additional Unit 3 topics to be covered in future sessions include: - Session 4: Connecting rates of change in polar functions. - Session 7: Modeling periodic functions from verbal descriptions. - Session 8: Rewriting trigonometric expressions and solving trigonometric equations.
Definition of a Periodic Function: - A periodic function is identified when, as input values increase, the output values demonstrate a repeating pattern over successive, equal-length intervals.
Real-World Examples of Periodic Behavior: - School Year Calendar: Teachers and students follow a repeating 12-month cycle. They are busy for 10 months and relatively relaxed with vacation for 2 months. This pattern repeats as 10 months of work followed by 2 months of free time. - Retail Sales: Many retail stores experience a peak in sales during November and December annually. This pattern repeats from year to year.

Characteristics of Sinusoidal Functions

Sinusoidal Categories: The primary sinusoidal functions are sine and cosine.
Graph Lifespan: While graphs typically show only one period or cycle, these functions repeat infinitely to the left and right.
Behavioral Patterns (Un-transformed): - Sine ( $\sin(\theta)$ ): Starts at zero (the midline), increases to a maximum of $1$ , decreases back through the midline to a minimum of $-1$ , and returns to the midline. - Cosine ( $\cos(\theta)$ ): Starts at its maximum value of $1$ (at input value zero), decreases to a minimum, and increases back to its maximum to complete one cycle.
Relationship Between Sine and Cosine: They are translations of one another. - Sine is cosine shifted to the right by $\frac{\pi}{2}$ . - Cosine is sine shifted to the left by $\frac{\pi}{2}$ . - Because of this, it is possible to use either function to model the same data set, depending on where the user chooses to start the cycle.

General Transformations of Sinusoidal Functions

Mathematical Model: $f(x) = a \sin(b(x+c)) + d$ or $f(x) = a \cos(b(x+c)) + d$ .
Vertical Transformations (Outside the function): - Vertical Dilation ( $a$ ): Represented by the value multiplied by the front. - Amplitude: Defined as the absolute value of $a$ ( $|a|$ ). It is the distance from the midline to the maximum or the distance from the midline to the minimum. It can also be calculated as half the distance from the minimum to the maximum: $\frac{\text{max} - \text{min}}{2}$ . - Vertical Translation ( $d$ ): Represented by added/subtracted values outside the argument. It shifts the graph vertically and establishes the midline.
Horizontal Transformations (Inside the function): - Horizontal Dilation ( $b$ ): Changes the length of the cycle by a factor of $\frac{1}{|b|}$ . - When $b = 1$ , the period of sine and cosine is $2\pi$ . - When $b = 2$ , two cycles must fit into the space of $2\pi$ , making the period for each cycle $\pi$ . - Horizontal Translation ( $c$ ): Also called the Phase Shift. The shift is by $-c$ units (the opposite direction of the sign).
Transformation Modalities: - Dilations ( $a$ and $b$ ) are multiplicative transformations. - Translations ( $c$ and $d$ ) are additive transformations.

Modeling from a Table of Values (Example 1)

Problem Context: Determining function parameters from a data table to find the best approximation for the constant $b$ in $f(x) = a \sin(b(x+c)) + d$ .
Step 1: Identifying the Period: - Data points: At input $4.5$ , output is $7.02$ ; at input $13.5$ , output is $7.04$ . - Both outputs represent the maximum. The distance between them is one full cycle. - $\text{Period} = 13.5 - 4.5 = 9$ .
Step 2: Calculating $b$ : - The relationship between period and $b$ is $\text{Period} = \frac{2\pi}{|b|}$ . - If $b$ is positive: $b = \frac{2\pi}{\text{Period}} = \frac{2\pi}{9}$ . - Approximation: $\frac{2 \times 3.14}{9} \approx \frac{6.28}{9} \approx 0.7$ . - Note: The transcript mentions "1.7" as the approximation for answer choice A, though based on the math provided ( $6.28/9$ ), it would be approximately $0.7$ .

Spin-off: Finding Amplitude and Midline (Example 1 Extended)

Data Analysis: - Maximum: Approximately $7.03$ (averaged or selected from $7.02$ and $7.04$ ). - Minimum: Found in the table at input $9$ with an output of $5.01$ .
Calculating Amplitude: - $\text{Amplitude} = \frac{7.03 - 5.01}{2} = \frac{2.02}{2} = 1.01$ . - A valid approximation for amplitude is $1$ .
Calculating Midline: - The midline is the average of the extrema. - $\text{Midline} = \frac{7.03 + 5.01}{2} = \frac{12.04}{2} = 6.02$ . - Midline equation: $y \approx 6.02$ .

Comprehensive Clock Hand Model (Example 2)

Scenario: A circular clock face with integers 1 to 12. At midnight ( $t = 0$ ), the hour hand points to 12. We model the height ( $y$ ) of the tip of the hour hand above the floor in inches.
Data Table Provided: - $t = 0$ , $y = 78$ . - $t = 3$ , $y = 70$ . - $t = 6$ , $y = 62$ . - $t = 9$ , $y = 70$ . - $t = 12$ , $y = 78$ .
Determining Parameters: - Maximum: $78$ inches (occurs at midnight/noon). - Minimum: $62$ inches (occurs at 6:00). - Midline: $\frac{78 + 62}{2} = 70$ . - Amplitude: $78 - 70 = 8$ . - Period: The time from max to max is $12 - 0 = 12$ hours. - Value of $b$ : $b = \frac{2\pi}{12} = \frac{\pi}{6}$ .
Choosing the Function (Negative Sine vs. Positive Cosine): - If a phase shift of $3$ units to the right is applied ( $c = -3$ ): - At $t = 3$ , the value is at the midline ( $70$ ). - Moving from $t = 3$ toward $t = 6$ , the graph decreases to the minimum. - A standard sine graph usually increases from its midline starting point. - Therefore, a negative sine function is required: $y(t) = -8 \sin(\frac{\pi}{6}(t - 3)) + 70$ .

The Six Trigonometric Functions and Tangent Analysis

Tangent Function ( $\tan(\theta)$ ): - Identity: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ . - Zeros (x-intercepts): Tangent equals zero wherever the numerator ( $\sin(\theta)$ ) equals zero. This occurs at $0, \pi, 2\pi, \dots$ . - Vertical Asymptotes: Tangent has asymptotes wherever the denominator ( $\cos(\theta)$ ) equals zero (provided the numerator is not also zero). This occurs at $\frac{\pi}{2}, \frac{3\pi}{2}, \dots$ .
Reciprocal Functions: - Cosecant ( $\csc(\theta)$ ): Defined as $\frac{1}{\sin(\theta)}$ . - Zeros of sine ( $0, \pi, 2\pi$ ) become vertical asymptotes for cosecant. - Since the numerator is $1$ , the graph of cosecant never equals zero. - Secant ( $\sec(\theta)$ ): Defined as $\frac{1}{\cos(\theta)}$ . - Cotangent ( $\cot(\theta)$ ): Defined as $\frac{1}{\tan(\theta)}$ or $\frac{\cos(\theta)}{\sin(\theta)}$ .

Vertical Asymptotes for Transformed Functions (Example 3)

Function: $h(\theta) = \tan(3\theta) + 1$ .
Effect of Vertical Translation: The " $+1$ " shifts the graph vertically but does not change the x-values where vertical asymptotes exist.
Method 1: Sine/Cosine Identity: - Identify where the denominator of the tangent function is zero: $\cos(3\theta) = 0$ . - Standard cosine zeros: $\dots, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \dots$ . - Set the argument equal to these values: $3\theta = \dots, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \dots$ . - Solve for $\theta$ : $\theta = \dots, -\frac{\pi}{6}, \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \dots$ .
Method 2: Period and Transformation Formula: - General asymptotes for $\tan(\theta)$ occur at $\theta = \frac{\pi}{2} + \pi k$ , where $k$ is an integer. - Given $b = 3$ , there are three cycles in the space of one original cycle. The input must be scaled by $\frac{1}{3}$ . - Transformed asymptotes: $\theta = \frac{1}{3} (\frac{\pi}{2} + \pi k) = \frac{\pi}{6} + \frac{\pi}{3} k$ .
Conclusion: The vertical asymptotes occur at $\theta = \frac{\pi}{6} + \frac{\pi}{3} k$ for integer $k$ .

Sinusoidal functions, primarily sine and cosine, demonstrate periodic behavior, characterized by repeating patterns over equal-length intervals.
The amplitude of sinusoidal functions indicates the distance from the midline to the maximum or minimum values, calculated as half the distance between these extrema.
Transformations of sinusoidal functions include vertical and horizontal shifts, dilations, and translations, encapsulated in the general mathematical model $f(x) = a ext{sin}(b(x+c)) + d$ .
Real-world periodic phenomena can be modeled using sinusoidal functions, with practical examples including the school year calendar and retail sales fluctuations.
The relationship between sine and cosine functions indicates they are translations of each other by $rac{ ext{π}}{2}$ , facilitating flexibility in modeling periodic data depending on chosen cycle initiations.