Exponents, Logarithms and their Applications — The exponential function and its applications

About this lesson

This unit introduces the exponential function f(x) = a^x, where a > 0 and a ≠ 1, and explores its graphical representation, properties, and applications. It begins with a real-world example of bacteria cell division to motivate exponential growth, then defines the exponential function and distinguishes it from algebraic functions. Students learn to identify exponential functions, draw their graphs, and analyze properties such as domain, range, monotonicity (increasing for a > 1, decreasing for 0 < a < 1), one-to-one nature, and the point (0,1). The unit also covers transformations of exponential graphs and applications to compound interest, population growth, and decay (e.g., car depreciation, medicine elimination). By the end, students should be able to recognize exponential functions, graph them, determine their properties, and solve applied problems involving growth and decay using formulas like A = P(1 + r/n)^(nt) and f(t) = a(1 ± r)^t.