Math Problem Statement

Which graph represents exponential decay?

Solution

To determine which graph represents exponential decay, let's quickly define it: Exponential decay is when a function decreases rapidly at first, then levels off toward zero as xx increases. This happens for functions of the form y=aekxy = a e^{-kx}, where a>0a > 0 and k>0k > 0, or more generally for forms like y=a(1b)xy = a \left(\frac{1}{b}\right)^x where 0<b<10 < b < 1.

Now, let's analyze the graphs in your image:

  1. Graph 1 (Top graph): This is a linear graph, not exponential, as it shows a straight line.
  2. Graph 2 (Second from the top): This graph increases slowly, indicating it might represent exponential growth, not decay.
  3. Graph 3 (Third graph): This graph starts high and decreases rapidly, flattening out as xx increases. This matches the behavior of exponential decay.
  4. Graph 4 (Bottom graph): This is also an increasing curve, suggesting it represents exponential growth.

Therefore, Graph 3 (third from the top) represents exponential decay.

Would you like further details or explanations?

Here are 5 questions to deepen your understanding:

  1. What is the general form of an exponential decay function?
  2. How can you differentiate between exponential growth and decay by looking at their graphs?
  3. What are the real-world applications of exponential decay?
  4. Can exponential decay functions ever reach zero?
  5. How does the rate of decay change as the value of the exponent increases?

Tip: Exponential decay is characterized by a base between 0 and 1, while growth has a base greater than 1.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Graph Analysis

Formulas

y = a e^{-kx}, a > 0, k > 0
y = a (1/b)^x, 0 < b < 1

Theorems

Exponential Decay

Suitable Grade Level

Grades 9-12