Recording-2025-02-14T15:22:32.154Z

Understanding Graphs of Functions

• Questions 13 through 19 will require sketches.

• For sketches, identify key components:

  • Label points a, b, and d.

  • Describe movement of blood as a metaphor for the function.

  • Points can still earn partial credit, even with sketch errors.

Graphing Functions

Basic Graphing Guidelines

• Range and amplitude adjustments:

  • Normal oscillation from 0 to 2π.

  • New unit oscillation can shift; example given is from 1 to 3 due to an upward shift.

• For cosine functions, start graph at the maximum amplitude:

  • Example with amplitude being shifted up results in starting point at 3.

Amplitude and Range

• Concept of amplitude: distance from the midline to peak.

  • Example given: amplitude of 1 causes points to range between 1 and 3.

• Shifts due to a, b, and d affect the graph's behavior.

  • D determines vertical shifts, which directly affect the height of the graph.

Adjustments to Amplitude, Period and Phase Shift

Analyzing Constants a, b, d

• a affects amplitude:

  • Example given is a = ( \frac{1}{2} ): amplitude becomes half, providing visual changes in height.

• b influences period and is calculated through the reciprocal relation:

  • When b = ( \frac{1}{2} ), new period calculation is 4π.

Shifts and Movements

• d determines vertical shifts:

  • Example: d = 1 moves the graph upward by one unit, adjusting the minimum and maximum values.

• Amplitude work example shows small changes:

  • After adding d of 1, adjust amplitude to find new peaks (1.5) and lows (0.5).

Key Graphing Concepts

Cosine Function Analysis

• Varies based on graph orientation:

  • Negative amplitude indicates inversion.

  • Absolute value of amplitude sets maximum possible height.

• Determining new period for a given fraction for b:

  • Ensure proper steps in calculations for timing and oscillation.

Utilizing Homework and Practice

• Students will sketch and graph functions as practice.

• Encouragement to complete additional exercises for comfort with graphing.

Review & Overall Concepts

Importance of Understanding Behavior

• Recognizing sine and cosine functions:

  • Identify periodic behaviors by starting points (max/min).

  • Connections between amplitude, range, and translation (up/down shifts).

• Knowing how to translate graphical information into function equations.

  • Use periods & decompositions for sine or cosine based on visual data from graphs.

Conclusion and Preparation for Exam

• No calculator is allowed on the main graphing portion of the test.

• The test will have two separate parts, both requiring clear demonstration of understanding in sketching and function behavior.