Eksponen & Logaritma Bagian 2 - Fungsi Eksponen - Pertumbuhan & Peluruhan Kelas X Kurikulum Merdeka

Introduction

This tutorial will guide you through the concepts of exponential functions as covered in the video "Eksponen & Logaritma Bagian 2 - Fungsi Eksponen - Pertumbuhan & Peluruhan." You will learn about the definition of exponential functions, their general form, graphical representation, and practical applications related to growth and decay. Whether you're studying for class or just looking to reinforce your understanding of these mathematical concepts, this guide will provide you with clear and actionable steps.

Step 1: Understand Exponential Functions

• Definition: An exponential function is a mathematical function of the form f(x) = a * b^x, where:
  • a is a constant (the initial value),
  • b is the base (a positive number),
  • x is the exponent (the variable).
• Characteristics:
  • If b > 1, the function represents exponential growth.
  • If 0 < b < 1, the function represents exponential decay.

Step 2: Explore the General Form of Exponential Functions

• The general form can be expressed as:
  • f(x) = a * b^x
• Key components:
  • Initial Value (a): This value determines where the graph starts on the y-axis.
  • Base (b): The growth or decay factor which influences the steepness of the curve.

Step 3: Analyze the Graph of Exponential Functions

• Growth Graph:
  • Starts low and increases rapidly.
  • Example: f(x) = 2 * 3^x shows a steep upward trend.
• Decay Graph:
  • Starts high and decreases towards zero.
  • Example: f(x) = 5 * (1/2)^x illustrates a gradual decline.
• Tips for Graphing:
  • Plot key points by substituting values for x.
  • Identify intercepts and behavior as x approaches positive or negative infinity.

Step 4: Solve Practical Examples

• Example 1: Find f(2) for f(x) = 3 * 4^x.
  • Substitute x with 2: f(2) = 3 * 4^2 = 3 * 16 = 48.
• Example 2: Identify the value of f(0).
  • For any exponential function, f(0) = a (the initial value).
• Example 3: Graph the function f(x) = 2 * 5^x and identify the growth rate.

Step 5: Apply Exponential Functions to Real-World Scenarios

• Exponential Growth:

  • Used in modeling populations, investments, and other scenarios where quantities increase rapidly over time.
  • Example: Population growth can be modeled by f(t) = P₀ * e^(rt), where P₀ is the initial population, r is the growth rate, and t is time.

• Exponential Decay:

  • Useful in scenarios such as radioactive decay or depreciation of assets.
  • Example: The decay of a substance can be modeled by f(t) = N₀ * e^(-kt), where N₀ is the initial amount and k is the decay constant.

Conclusion

In this tutorial, we covered the essential aspects of exponential functions, including their definition, general form, graphical representation, and practical applications in growth and decay scenarios. By understanding these concepts, you will be better equipped to tackle problems involving exponential functions in your studies. For further practice, consider working on additional examples or exploring more complex applications in real-world contexts.