Chapter Outline
In Unit 5, “Exponential Functions,” we explore the dynamic nature of exponential growth and decay. The unit begins by identifying and graphing exponential growth functions from various data sources, emphasizing their rate of change, domain, and range. Students will learn to model real-world scenarios with these functions, understanding constraints that affect their applications. The exploration continues with exponential decay, paralleling the growth section with similar objectives but focusing on decreasing values. Additionally, the unit covers vertical transformations of exponential functions and delves into geometric sequences, examining them as discrete exponential functions with restricted domains. This comprehensive approach not only enhances understanding of exponential functions but also equips students with practical modeling skills.
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5.2
Exponential Growth Functions
Learning Objectives
By the end of this section, you will be able to:
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analyze exponential growth functions to identify key features such as the base and initial value, and understand how these influence the function’s behavior.
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determine the rate of change, domain, and range of exponential growth functions, enabling me to describe their growth limits and rates accurately.
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formulate exponential growth functions from real-world data or scenarios, accurately modeling exponential increases and predicting future outcomes.
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graph exponential growth functions and interpret their properties to gain insights into their visual representations and underlying behaviors.
5.3
Exponential Decay Functions
Learning Objectives
By the end of this section, you will be able to:
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identify the characteristics of exponential decay functions including the base, initial value, and the nature of the decay over time.
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determine the rate of change, domain, and range of exponential decay functions and understand how these features define the function’s behavior.
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write mathematical models in the form of exponential decay functions based on given data or scenarios.
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graph exponential decay functions and analyze their properties to understand the graphical representation of decay over time.
5.4
Introduction to Sequences
Learning Objectives
By the end of this section, you will be able to:
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analyze patterns to determine if a sequence is arithmetic, geometric, or neither, and identify the characteristics of each type.
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calculate a specific term of a sequence using its defining properties, such as common differences in arithmetic sequences or common ratios in geometric sequences.
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create a recursive formula for a sequence, enabling me to calculate any term based on its predecessors.
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apply my understanding of sequences to solve problems and create mathematical models that reflect the progression of these sequences.
5.5
Arithmetic Sequences
Learning Objectives
By the end of this section, you will be able to:
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analyze patterns to determine if a sequence is arithmetic or geometric and identify the common differences or ratios.
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calculate specific terms of a sequence using the properties of arithmetic and geometric sequences.
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create recursive formulas for sequences, allowing me to establish relationships between successive terms.
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apply my understanding of sequences to solve problems and create mathematical models that reflect the progression of these sequences.
5.6
Geometric Sequences
Learning Objectives
By the end of this section, you will be able to:
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analyze geometric sequences to determine their characteristics, including the common ratio and specific properties that define their behavior.
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calculate the common ratio of a geometric sequence and identify any domain restrictions that apply.
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compare and contrast explicit and recursive formulas used to describe geometric sequences and understand their applications.
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graph geometric sequences and interpret these graphs to understand how sequences progress visually.