Graphs of trigonometric functions are the visual representation of sine, cosine, and tangent. These functions oscillate periodically, creating wave-like patterns that repeat at regular intervals. Understanding their graphs is crucial for modeling real-world phenomena like sound waves and seasonal changes.

Transformations allow us to modify these graphs, changing their amplitude, period, and position. By applying shifts, stretches, and reflections, we can create complex functions that accurately model periodic events. This ability to manipulate trigonometric graphs is essential for solving real-world problems in science and engineering.

Graphing trigonometric functions

Characteristics of sine, cosine, and tangent functions

• The sine function, f(x) = sin x, is a periodic function with a period of 2π, an amplitude of 1, and a range of [-1, 1]
  • The graph oscillates smoothly between -1 and 1, crossing the x-axis at multiples of π (0, π, 2π, etc.)
• The cosine function, f(x) = cos x, is a periodic function with a period of 2π, an amplitude of 1, and a range of [-1, 1]
  • The graph oscillates smoothly between -1 and 1, reaching a maximum value of 1 at multiples of 2π (0, 2π, 4π, etc.) and a minimum value of -1 at odd multiples of π (π, 3π, 5π, etc.)
• The tangent function, f(x) = tan x, is a periodic function with a period of π, an undefined amplitude, and a range of (-∞, ∞)
  • The graph has vertical asymptotes at odd multiples of π/2 (π/2, 3π/2, 5π/2, etc.) and crosses the x-axis at multiples of π (0, π, 2π, etc.)

Key features and midline of trigonometric functions

• Key features of trigonometric functions include:
  • Amplitude: the maximum displacement from the midline
  • Period: the length of one complete cycle
  • Phase shift: the horizontal translation of the graph
  • Vertical shift: the vertical translation of the graph
• The midline of a trigonometric function is the horizontal line that passes through the center of the graph
  • Typically represented by the equation y = 0 for the standard sine, cosine, and tangent functions
  • Functions with a vertical shift will have a midline at y = k, where k is the constant term added to the function

Transformations of trigonometric graphs

Types of transformations

• Transformations of trigonometric functions include translations (shifts), reflections, stretches, and compressions
  • These transformations alter the appearance and key features of the graph without changing the fundamental shape
• Vertical translations shift the graph up or down by adding a constant term to the function, e.g., f(x) = sin x + k
  • Positive values of k shift the graph up, while negative values shift it down
• Horizontal translations shift the graph left or right by adding a constant term to the input variable, e.g., f(x) = sin(x - h)
  • Positive values of h shift the graph to the right, while negative values shift it to the left

Reflections, stretches, and compressions

• Reflections flip the graph across the x-axis or y-axis
  • Reflecting across the x-axis is achieved by negating the function, e.g., f(x) = -sin x
  • Reflecting across the y-axis is achieved by negating the input variable, e.g., f(x) = sin(-x)
• Vertical stretches and compressions change the amplitude of the function by multiplying the function by a constant term, e.g., f(x) = a sin x
  • Values of |a| > 1 stretch the graph vertically, while values of |a| < 1 compress it vertically
• Horizontal stretches and compressions change the period of the function by dividing the input variable by a constant term, e.g., f(x) = sin(x/b)
  • Values of |b| > 1 compress the graph horizontally (decreasing the period), while values of |b| < 1 stretch it horizontally (increasing the period)

Combining transformations

• Multiple transformations can be applied to a single trigonometric function, and the order in which they are applied matters
• Generally, the order of transformations is as follows:
  1. Reflections
  2. Stretches/compressions
  3. Horizontal translations
  4. Vertical translations
• Combining transformations allows for the creation of complex trigonometric functions that model various periodic phenomena

Modeling periodic phenomena

Characteristics of periodic phenomena

• Periodic phenomena are events or behaviors that repeat at regular intervals
  • Examples include sound waves, tides, and seasonal changes (temperature, daylight hours)
• Trigonometric functions are well-suited for modeling these phenomena due to their periodic nature
• To model periodic phenomena, identify the key characteristics of the event:
  • Amplitude: maximum displacement from the midline
  • Period: time required for one complete cycle
  • Phase shift: horizontal translation
  • Vertical shift: vertical translation

Choosing the appropriate trigonometric function

• Determine the appropriate trigonometric function (sine, cosine, or tangent) based on the characteristics of the phenomenon and the starting point of the cycle
  • Use a sine function if the phenomenon starts at the midline (e.g., a pendulum starting at its equilibrium position)
  • Use a cosine function if the phenomenon starts at a maximum or minimum value (e.g., the height of a Ferris wheel car starting at the top or bottom of the wheel)
• Apply the necessary transformations to the chosen trigonometric function to match the key characteristics of the phenomenon
  • Change the amplitude, period, phase shift, and vertical shift as needed
• Interpret the resulting model in the context of the phenomenon, and use it to make predictions or draw conclusions about the behavior of the event over time

Amplitude, period, and phase shift

Finding the amplitude

• To find the amplitude of a trigonometric function, identify the coefficient of the function (the value that multiplies the sine, cosine, or tangent term)
  • The absolute value of this coefficient represents the amplitude
  • Example: In the function f(x) = 3 sin x, the amplitude is |3| = 3

Calculating the period

• To find the period of a trigonometric function, identify the coefficient of the input variable within the function (the value that divides the variable inside the parentheses)
  • The period is calculated as 2π divided by the absolute value of this coefficient
  • Example: In the function f(x) = cos(2x), the period is 2π / |2| = π

Determining the phase shift

• To find the phase shift of a trigonometric function, identify the constant term added to or subtracted from the input variable within the function
  • This value represents the horizontal translation of the graph, with positive values shifting the graph to the left and negative values shifting it to the right
  • Example: In the function f(x) = sin(x - π/4), the phase shift is π/4, meaning the graph is shifted to the right by π/4 units

Solving problems involving amplitude, period, and phase shift

• When solving problems involving amplitude, period, and phase shift, use the given information to set up an equation representing the trigonometric function
  • Manipulate the equation to isolate the desired variable and solve for its value
• In some cases, you may need to use the properties of trigonometric functions, such as the Pythagorean identity ($\sin^2x + \cos^2x = 1$) or the periodicity of the functions, to simplify the equation or determine additional information about the function
• Verify your solution by graphing the trigonometric function with the calculated amplitude, period, and phase shift
  • Ensure that the graph matches the given information or constraints in the problem