Characteristics of Sinusoidal Functions

Ms. Madamba's AP Precalculus - Unit 3 Review: Characteristics of Sinusoidal Functions

Overview

• Focus: Understanding key characteristics of sinusoidal functions including amplitude, midline, period, and phase shift.
• Tasks Provided:
    - Part 1: Analyze given sinusoidal functions.
    - Part 2: Construct sinusoidal functions based on specified characteristics.
    - Part 3: Answer questions related to provided graphs.

Part 1: Given Sinusoidal Functions

Task

• Determine the amplitude, midline, period, and phase shift of the given sinusoidal functions.
• For any phase shift, state whether it is to the left or the right. If no phase shift exists, state none.
• Show work for computing the period.

Function Analysis

1. Function: $y = 3 \, ext{cos}(2 ext{π}(x + \frac{4}{3}))$
      - Amplitude: $3$
      - Period:
        - Defined as $\frac{2 ext{π}}{b}$ where $b = 2 ext{π}$, thus:
   $ext{Period} = \frac{2 ext{π}}{2 ext{π}} = 1$
      - Midline: $y = 0$
      - Phase Shift:
        - Shift to the left by $\frac{4}{3}$.

2. Function: $y = 7 \, ext{sin}(x) + 2$
      - Amplitude: $7$
      - Midline: $y = 2$
      - Period:
        - Since $b = 1$,
   $ext{Period} = \frac{2 ext{π}}{1} = 2 ext{π}$
      - Phase Shift: none.

3. Function: $y = ext{cos}\bigg(x - 5\bigg) + 0$
      - Amplitude: $1$
      - Midline: $y = 0$
      - Period:
        - $b = 1$, hence
   $ext{Period} = 2 ext{π}$
      - Phase Shift: to the right by $5$.

4. Function: $y = 9 \, ext{sin}\bigg(x + \frac{ ext{π}}{8}\bigg)$
      - Amplitude: $9$
      - Midline: $y = 0$
      - Period:
        - $b = 1$, thus
   $ext{Period} = 2 ext{π}$
      - Phase Shift: to the left by $\frac{ ext{π}}{8}$.

Part 2: Constructing Sinusoidal Functions

Task

• Write sinusoidal functions based on given characteristics and calculate the $b$ value when required.

1. Write a sine function with:
      - Amplitude: $6$, Midline: $y = 5$, Period: $12$.
      - Calculation:
        - $T = 12$ leads to
   $b = \frac{2 ext{π}}{T} = \frac{2 ext{π}}{12} = \frac{ ext{π}}{6}$
      - Function: $f(x) = 6 \, ext{sin}\bigg(\frac{ ext{π}}{6}x\bigg) + 5$.

2. Write a cosine function with:
      - Amplitude: $6$, Midline: $y = 7$, Period: $8$, horizontal shift to the right.
      - Calculation:
        - $T = 8$ leads to
   $b = \frac{2 ext{π}}{T} = \frac{2 ext{π}}{8} = \frac{ ext{π}}{4}$
      - Function: $f(x) = 6 \, ext{cos}\bigg(\frac{ ext{π}}{4}(x - c)\bigg) + 7$ where $c$ is the phase shift.

3. Write a sine function with:
      - Amplitude: $4$, Midline: $y = 3$, Period: $5 ext{π}$, horizontal shift 7 to the left.
      - Calculation:
        - $T = 5 ext{π}$ leads to
   $b = \frac{2 ext{π}}{5 ext{π}} = \frac{2}{5}$
      - Function: $f(x) = 4 \, ext{sin}\bigg(\frac{2}{5}(x + 7)\bigg) + 3$.

Part 3: Graph Analysis

Task

• Analyze graphs to determine amplitude, midline, period, and write corresponding sinusoidal functions.

1. Question #9:
      - Graph Characteristics:
        - Amplitude: $2$
        - Midline: $y = 0$
        - Period: $8$
      - Function without phase shift:
   $f(x) = 2 \, ext{cos}(x)$.

2. Question #10:
      - Graph Characteristics:
        - Amplitude: $2$
        - Midline: $y = -3$
        - Period: $3 ext{π}$
      - Function with phase shift:
   $f(x) = -2 \, ext{sin}\bigg(\frac{3}{3}x\bigg) + 1$.

3. Question #11:
      - Graph Characteristics:
        - Amplitude: $1$
        - Midline: $y = -3$
        - Period: $16$
      - Function with phase shift:
   $f(x) = -1 \, ext{sin}\bigg(x - 2\bigg) - 3$ (several equivalent forms possible).

Summary

• This review emphasizes key characteristics of sinusoidal functions essential for the understanding of their behavior in mathematical contexts. Understanding how to manipulate and read these functions is critical for advanced mathematics.