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.