Graphing Sine and Cosine Trig Functions With Transformations, Phase Shifts, Period - Domain & Range

TL;DR

Learn how to graph sine and cosine functions with different amplitudes, periods, and phase shifts.

Transcript

in this video we're going to focus on how to graph sine and cosine functions particularly with horizontal phase shifts but we're going to start with some basic structures some basic equations and then we're going to make it progressively harder so let's say if we want to graph let's start with the very basic y equals sine X the amplitude is 1 and f... Read More

Key Insights

• 🫥 The amplitude of a sine or cosine function affects the vertical distance of the wave from the center line.
• 👋 The period of a sine or cosine function determines the length of one complete cycle or wave.
• ☺️ Phase shifts determine where the wave starts horizontally along the x-axis.
• 🧑‍🏭 Sine and cosine functions have specific equations, with factors for amplitude, period, phase shift, and vertical shift.
• 🔙 The generic equation for a sine or cosine function is y = a*sin(Bx + C) + D, where a is the amplitude, B determines the period, C represents the phase shift, and D is the vertical shift.
• 🤩 Graphing a sine or cosine function involves plotting key points based on the amplitude, period, and phase shift, and connecting them to form the waveform.
• 🫥 The vertical shift determines the position of the center line of the graph.
• 🧡 The range of a sine or cosine function depends on the amplitude and vertical shift, while the domain is typically unlimited unless restricted by the phase shift.

Questions & Answers

Q: What is the amplitude of a sine or cosine function?

The amplitude determines how high or low the wave extends from the center. It represents the maximum distance the wave reaches from the center line.

Q: How do you calculate the period of a sine or cosine function?

The period is the length it takes for the wave to complete one full cycle. It can be calculated using the formula: period = 2π / B, where B is the number in front of X in the equation.

Q: How does a phase shift affect the graph of a sine or cosine function?

A phase shift determines where the wave starts horizontally along the x-axis. It can be calculated by setting the inside of the function equal to zero and solving for X.

Q: Can you restrict the domain of a sine or cosine graph?

By setting specific values for the phase shift, you can restrict the domain of the graph. Otherwise, the domain typically extends from negative infinity to infinity.

Summary & Key Takeaways

• Sine and cosine functions have specific patterns and characteristics, such as amplitude, period, and phase shifts.

• The amplitude of a sine or cosine function determines how high or low the wave extends from the center.

• The period of a sine or cosine function determines how long it takes for the wave to complete one full cycle.

• Phase shifts affect where the wave starts horizontally along the x-axis.