#Sinusoidal Functions: Your Ultimate Review 🚀

Hey there, future AP Pre-Calculus master! Let's dive into the world of sinusoidal functions. Think of this as your pre-exam power-up – everything you need, nothing you don't. Let's get started!

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Understanding Sinusoidal Functions

#What are Sinusoidal Functions?

• A sinusoidal function is any function that looks like a sine or cosine curve – oscillating and periodic. Both sine and cosine functions are sinusoidal, and so are their transformations.

• Key Idea: The cosine function is just a transformed sine function:

  $$cos(θ) = sin(θ + \frac{π}{2})$$

  This means the cosine curve is the sine curve shifted to the left by $\frac{π}{2}$ units. Think of it as a horizontal slide! ➡️

  Image courtesy of Medium.

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Characteristics of Sinusoidal Functions

These characteristics are super important! Knowing them is like having a secret code to unlock any sinusoidal graph.

#Period (T)

• The period is the length of one complete cycle of the function. It's how long it takes for the wave to repeat itself. 🌊
• For standard sine and cosine, the period is $2π$. But remember, transformations can change this!
• How to find it: Measure the distance between any point on the graph and the next time the graph reaches that same y-value going in the same direction.

#Frequency (f)

• The frequency is how many cycles occur in one unit of time. It's the reciprocal of the period.
• Formula: $$f = \frac{1}{T}$$
• For sine and cosine, the frequency is $\frac{1}{2π}$.

#Midline (k)

• The midline is the horizontal line that cuts the wave in half. It's the average y-value of the function.
• How to find it: Average the maximum and minimum y-values: $$k = \frac{y_{max} + y_{min}}{2}$$
• For standard sine and cosine, the midline is $y = 0$.

#Amplitude

• The amplitude is the vertical distance from the midline to the maximum (or minimum) value of the function. It's always a positive value. 📏

• How to find it: $$Amplitude = \frac{y_{max} - y_{min}}{2}$$

• For standard sine and cosine, the amplitude is 1. ### Symmetry

• Odd Symmetry: Symmetric about the origin. If you rotate the graph 180 degrees, it looks the same. Mathematically, $f(-x) = -f(x)$. The sine function has odd symmetry.

• Even Symmetry: Symmetric about the y-axis. If you reflect the graph over the y-axis, it looks the same. Mathematically, $f(-x) = f(x)$. The cosine function has even symmetry.

#Oscillation and Concavity

• Oscillation: Sinusoidal functions move up and down between two values, creating a wave-like pattern.

• Concave Up: The curve looks like a smile. 😊

• Concave Down: The curve looks like a frown. 🙁

  Image courtesy of MathLibreTexts.

  In the graph above:

  • Concave up: x=1 to x=3 and x=5 to x=7
  • Concave down: x=-1 to x=1, x=3 to x=5, and x=7 to x=9