Determine an equation for each graph.

Verified step by step guidance

1. insert step 1: Identify the type of trigonometric function that best fits the graph, such as sine, cosine, tangent, etc.
2. insert step 2: Determine the amplitude of the graph by measuring the vertical distance from the midline to the peak or trough.
3. insert step 3: Find the period of the graph by identifying the horizontal length of one complete cycle.
4. insert step 4: Determine any phase shift by observing how much the graph is shifted horizontally from the standard position.
5. insert step 5: Identify any vertical shift by determining how much the midline of the graph is shifted up or down from the x-axis.

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting the sine, cosine, and tangent functions on a coordinate plane. Each function has a unique shape characterized by its amplitude, period, and phase shift. Understanding these properties is essential for accurately determining the equation that represents a given graph.

Amplitude and Period

Amplitude refers to the maximum height of a wave from its midline, while the period is the distance over which the function completes one full cycle. For sine and cosine functions, the standard amplitude is 1, and the period is 2π. Adjustments to these values in the function's equation affect the graph's appearance, making it crucial to identify them when determining the equation.

Phase Shift

Phase shift describes the horizontal displacement of a trigonometric graph from its standard position. It occurs when the function is modified by adding or subtracting a constant inside the function's argument. Recognizing the phase shift is vital for accurately writing the equation that corresponds to the graph, as it affects where the function starts on the x-axis.