Day 1 of 30
Exponential functions: f(x) = aˣ
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Learning Objective
Students will explore exponential functions and f(x) = aˣ through guided practice and application.
Welcome to Exponential and Logarithmic Thinking! This lesson focuses on exponential functions: f(x) = aˣ. Mathematics provides tools for modeling and analyzing complex systems.
Exponential functions have the form f(x) = a·bˣ (or f(x) = a·e^(kx)). For growth: b > 1 or k > 0; for decay: 0 < b < 1 or k < 0. They model population growth, radioactive decay, compound interest, and more. Key property: constant percent rate of change. The number e ≈ 2.718 is the natural base for continuous growth/decay!
As you engage with this material, consider both the theoretical foundations and practical applications. Think critically about how this concept builds on prior knowledge and where you might apply it beyond the classroom.
Challenge yourself to go beyond memorization—seek to understand the "why" behind the processes and principles.
A mathematical question requiring analysis and solution
The result obtained by solving a problem
Instructions
Materials Needed
- Notebook
- Calculator (if needed)
- Graph paper
Simplify: x³ · x⁴
Simplify: (x²)³
Simplify: x⁰
Teaching Tip
For advanced learners: Encourage deeper analysis and real-world connections. Consider extension activities that allow students to apply exponential functions: f(x) = aˣ to novel situations.