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### Overview

In this classroom resource from Desmos, students will build a visual understanding of amplitude, period, and phase shift with an introduction to trigonometric graphing. They can use this understanding to find models for given graphs of the sine function. The resource could also be used to help teach a lesson on trigonometric graphs. The resource includes sample student responses and a teacher's guide.

## UP:MA19.A2.35

### Vocabulary

• Trigonometric functions
• Periodic phenomena
• Amplitude
• Frequency
• Midline

### Knowledge

Students know:
• Key features of trigonometric functions (e.g., amplitude, frequency, and midline).
• Techniques for selecting functions to model periodic phenomena.

### Skills

Students are able to:
• Determine the amplitude, frequency, and midline of a trigonometric function.
• Develop a trigonometric function to model periodic phenomena.

### Understanding

Students understand that:
• Trigonometric functions are periodic and may be used to model certain periodic contextual phenomena.
• Amplitude, frequency, and midline are useful in determining the fit of the function used to model the phenomena.

## UP:MA19.A2.20

### Vocabulary

• Polynomial function
• Logarithmic function Trigonometric (sine and cosine) function
• Reciprocal function
• Period
• Midline
• Amplitude
• End Behavior
• Intervals
• Maximum
• Minimum
• Horizontal Asymptote
• Vertical Asymptote
• Inverse functions

### Knowledge

Techniques for graphing.

• Key features of graphs of functions.
• ### Skills

Students are able to:

• Identify the type of function from the symbolic representation.
• Manipulate expressions to reveal important features for identification in the function.
• Accurately graph any relationship.
• Find the inverse of a function algebraically and/or graphically.

### Understanding

Students understand that:

• Key features are different depending on the function.
• Identifying key features of functions aid in graphing and interpreting the function.
• A function and its inverse are reflections over the line y = x.

### Body

1. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions. a. Graph polynomial functions expressed symbolically, identifying zeros when suitable factorizations are available, and showing end behavior. b. Graph sine and cosine functions expressed symbolically, showing period, midline, and amplitude. c. Graph logarithmic functions expressed symbolically, showing intercepts and end behavior. d. Graph reciprocal functions expressed symbolically, identifying horizontal and vertical asymptotes. e. Graph square root and cube root functions expressed symbolically. f. Compare the graphs of inverse functions and the relationships between their key features, including but not limited to quadratic, square root, exponential, and logarithmic functions.

### Evidence of Student Attainment

Students: <ul><li>
Given a symbolic representation of a function (including polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, and radical), produce an accurate graph (by hand in simple cases and by technology in more complicated cases) and justify that the graph is an alternate representation of the symbolic function.</li></ul>

### CR Resource Type

Learning Activity

### Resource Provider

Desmos
Accessibility