8.2.1: Resources and Key Concepts

Prerequisite Topics

This page lists the prerequisite skills (all of which can be reviewed in CRC's Corequisite Codex) needed for this section that have not already been mentioned in any previous section. If you are enrolled in a course with a Support section, some (but definitely not all) of these topics might be covered or reviewed in the Support section of your course.

• Graphing Functions
  • Transformations: This section is entirely about applying non-rigid transformations (stretches, compressions, and reflections) to the base graphs of trigonometric functions. A prior understanding of how parameters alter a base graph is essential.

Videos

• Transformations
  • Trigonometric Graphs - Review of Graphs of Sine and Cosine Functions
  • Trigonometric Graphs - Review of Identifying the Equation for a Sinusoidal Function from a Graph
  • Graphs of the Other Trigonometric Functions - Graphing Tangent and Cotangent Functions
  • Graphs of the Other Trigonometric Functions - Identifying Equations of Trig Functions from Graphs

Definitions

• Transformation (of the graph of a function): An alteration of a graph's orientation, scale, and/or position through various mathematical operations.
• Rigid Transformation: A transformation that changes a graph's position (e.g., shifts) but keeps its original orientation and scale.
• Non-rigid Transformation: A transformation that keeps the graph's original center position but changes its scale (stretching or compressing) or orientation (reflecting).
• Parameter: A variable that influences the output or behavior of a function but is held constant within a specific context. In this section, \(A\) and \(B\) in \(y = A\text{trig}(Bx)\) are parameters.
• Key Numbers: The x-values of the initial, final, middle, and quarter-points of a single cycle of a trigonometric function's graph.
• Step Size: The horizontal distance between any two consecutive key numbers.

Theorems

• Vertical Scaling: The graph of \(f(x) = A \text{trig}(x)\) vertically scales the graph of \(y = \text{trig}(x)\) by a factor of \(|A|\). If \(|A| > 1\), it is a vertical stretch. If \(0 < |A| < 1\), it is a vertical compression. If \(A < 0\), the graph is also reflected about the horizontal axis.
• Amplitude of a Sinusoidal Function: The amplitude of \(f(x) = A\cos(x)\) or \(g(x) = A\sin(x)\) is \(|A|\).
• Period of a Trigonometric Function: If \(p\) is the natural period of a trigonometric function, then the period of \(y = A\text{trig}(Bx)\) is \(\frac{p}{|B|}\). For sine and cosine, the period is \(\frac{2\pi}{|B|}\). For tangent, the period is \(\frac{\pi}{|B|}\).
• Step Size Formula: The step size for graphing a trigonometric function is given by the formula \(\frac{\text{Period}}{4}\).

Common Mistakes

• Confusing Period and the Parameter B: The period of a function like \(y = \sin(Bx)\) is not \(B\). The period is calculated by dividing the natural period of the base function by \(|B|\).
• Using the Wrong Natural Period: The natural period of sine and cosine is \(2\pi\), while the natural period of tangent is \(\pi\). Using the wrong natural period when calculating the new period is a common error.
• Ignoring a Negative Coefficient on the Angle: When graphing a function like \(y = \tan(-4x)\), you must first use the Symmetry Identities to handle the negative. Since tangent is an odd function, \(y = \tan(-4x) = -\tan(4x)\). Graphing without this first step will lead to an incorrect orientation.
• Forgetting Tangent Has No Amplitude: The concept of amplitude applies to sinusoidal functions (sine and cosine) because they have maximum and minimum values. The tangent function is unbounded, so it does not have an amplitude.