11.3 Graphs of Trigonometric Functions
5 min read•july 31, 2024
Graphs of trigonometric functions are the visual representation of , , and . 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 , , 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
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- 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
- : the of the graph
- : 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:
- Reflections
- Stretches/compressions
- Horizontal translations
- 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 () 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
Key Terms to Review (21)
Amplitude: Amplitude is the maximum distance a wave or oscillating function reaches from its central position, typically measured from the equilibrium point to the peak (or trough) of the wave. This concept is essential in understanding various characteristics of waveforms and periodic functions, as it directly affects their height and overall shape. The amplitude of trigonometric functions reveals important information about their behavior and applications, particularly in relation to their graphical representations and real-world phenomena.
Cosine: Cosine is a trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It plays a crucial role in understanding periodic functions, as it describes how angles map onto the unit circle and is essential in graphing waveforms and solving triangles.
Frequency: Frequency refers to the number of occurrences of a repeating event per unit of time. In the context of waveforms, particularly trigonometric functions, frequency indicates how often a wave oscillates in a given time period and is often expressed in hertz (Hz). It plays a crucial role in understanding the behavior and characteristics of waves and signals, linking concepts in mathematics and real-world applications like sound, light, and mechanical vibrations.
Horizontal shift: A horizontal shift refers to the transformation of a graph where it is moved left or right along the x-axis. This change occurs when a constant is added to or subtracted from the input variable of a function, altering its original position while maintaining its shape. Understanding horizontal shifts is crucial as they can affect the behavior and properties of various types of functions, including their intercepts, asymptotes, and periodicity.
Horizontal translation: Horizontal translation refers to the shifting of a graph left or right along the x-axis without changing its shape or orientation. This transformation is significant because it alters the input values of a function, effectively adjusting the starting point of periodic behaviors, especially in trigonometric functions, such as sine and cosine waves.
Maxima: Maxima refers to the highest points on a graph of a function, representing local or global peaks where the function reaches its maximum value within a given interval. These points are crucial in understanding the behavior of trigonometric functions, as they indicate where the function achieves its greatest height and help identify key features like amplitude and oscillation patterns.
Minima: Minima refer to the lowest points on a graph of a function, where the value of the function reaches its minimum in a given interval. In the context of trigonometric functions, minima can be identified where the sine or cosine graphs reach their least value, impacting their overall shape and periodicity. Understanding minima is crucial for analyzing the behavior and characteristics of these functions, including amplitude and oscillation patterns.
Period: In trigonometric functions, the period is the length of one complete cycle of the function. This means that if you graph a trigonometric function, such as sine or cosine, the period represents how far you need to move along the x-axis before the graph starts repeating itself. The concept of period is crucial for understanding wave patterns, oscillations, and many real-world applications in physics and engineering.
Phase Shift: Phase shift refers to a horizontal translation of a periodic function, such as trigonometric functions, that alters the starting point of its cycle. It plays a vital role in understanding how these functions behave over the unit circle and how their graphs appear when shifted left or right. By applying a phase shift, the graph can be adjusted to model real-world scenarios more accurately, reflecting changes in timing or position.
Point plotting: Point plotting refers to the process of determining and marking specific points on a coordinate plane based on given mathematical functions or relationships. This technique is essential in visualizing graphs, especially for trigonometric functions, as it helps in understanding their behaviors, such as periodicity, amplitude, and phase shift. By accurately plotting points, one can create a clear representation of these functions and analyze their properties effectively.
Reference Angle: A reference angle is the smallest angle formed between the terminal side of an angle and the x-axis, measured in a positive direction. This concept is vital for determining the trigonometric function values for angles in different quadrants, as reference angles provide a way to simplify these calculations by relating them back to angles in the first quadrant.
Reflection: Reflection refers to the flipping or mirroring of a figure across a specific line, called the line of reflection, which acts as a mirror. This concept is vital in understanding how shapes can be transformed while maintaining their properties, making it applicable in various mathematical contexts such as symmetry, geometric transformations, and function analysis.
Sine: Sine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. It plays a crucial role in various mathematical contexts, particularly in the analysis of angles and their properties, as well as in the representation of periodic phenomena through wave functions.
Tangent: The tangent function is a fundamental trigonometric function that represents the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. It connects closely to angles on the unit circle, where it can be defined as the y-coordinate divided by the x-coordinate for any point on the circle. This relationship reveals how tangent behaves periodically, reflecting its unique graphical features and its role in various identities and proofs.
Unit circle: The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It serves as a fundamental tool in trigonometry, providing a visual way to understand the relationships between angles and the values of sine, cosine, and tangent. The unit circle connects the concepts of angles measured in both degrees and radians, while also being essential for graphing trigonometric functions and their properties.
Using symmetry: Using symmetry in mathematics refers to the property of a figure or graph being invariant under certain transformations, such as reflection, rotation, or translation. In the context of trigonometric functions, symmetry helps in understanding the behavior and characteristics of the graphs, making it easier to identify key features like amplitude, period, and phase shift. Recognizing these symmetries can simplify the graphing process and enhance overall comprehension of trigonometric functions.
Vertical Shift: A vertical shift refers to the movement of a graph up or down along the y-axis without altering its shape. This transformation is achieved by adding or subtracting a constant value to the output of a function, effectively translating the entire graph in a vertical direction. Such shifts can change the location of intercepts and influence the range of the function, while retaining the basic characteristics of the original graph.
X-intercepts: X-intercepts are the points where a graph crosses the x-axis, indicating the values of x for which the corresponding y-value is zero. These points are essential in understanding the behavior of functions, particularly quadratic and trigonometric functions. Identifying x-intercepts helps in solving equations and analyzing the roots of functions, making them a vital part of graphical analysis.
Y = a cos(bx + c) + d: This equation represents the general form of a cosine function, where 'a' indicates the amplitude, 'b' affects the period, 'c' is the phase shift, and 'd' represents the vertical shift. Understanding this equation helps in graphing trigonometric functions and recognizing their key characteristics such as maximum and minimum values, symmetry, and periodic behavior.
Y = a sin(bx + c) + d: The equation y = a sin(bx + c) + d represents a sine function that has been transformed in several ways, including amplitude, period, phase shift, and vertical shift. The variable 'a' affects the amplitude, which is the height of the wave; 'b' determines the frequency and period of the wave; 'c' shifts the graph horizontally, and 'd' shifts it vertically. These transformations allow for a wide variety of sine curves that can model real-world periodic phenomena.
Y-intercepts: Y-intercepts are the points where a graph intersects the y-axis, which indicates the value of a function when the input (x) is zero. Understanding y-intercepts is crucial when analyzing graphs, especially for trigonometric functions, as they help identify key characteristics like amplitude and periodicity. In the context of trigonometric graphs, y-intercepts can reveal important properties about the function's symmetry and transformation from the parent functions.