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For Aufgabe 1a, we need to match each of the given functions with the corresponding graph in the coordinate plane shown.

The functions provided are:
1. \( f(x) = 2^x \)
2. \( f(x) = 2 \cdot 0.8^x \)
3. \( f(x) = 2 \cdot 0.5^x \)
4. \( f(x) = 2 \cdot 1.25^x \)
5. \( f(x) = 2 \cdot 2^x \)

These are exponential functions, where each function has the form \( f(x) = c \cdot a^x \) with different values for the base \( a \).

### Step-by-Step Analysis of Each Function:
1. **\( f(x) = 2^x \)**: This is a standard exponential growth function with base 2. It should appear as a graph that increases rapidly as \( x \) becomes positive and decreases towards 0 as \( x \) becomes negative.

2. **\( f(x) = 2 \cdot 0.8^x \)**: The base is \( 0.8 \), which is less than 1. This means it's a decaying exponential function, gradually approaching zero as \( x \) increases. Since \( c = 2 \), the function starts at \( y = 2 \) when \( x = 0 \).

3. **\( f(x) = 2 \cdot 0.5^x \)**: With a base of \( 0.5 \), this function decays more quickly than \( f(x) = 2 \cdot 0.8^x \). It also has an initial value of 2 at \( x = 0 \).

4. **\( f(x) = 2 \cdot 1.25^x \)**: The base \( 1.25 \) is slightly above 1, so this function shows slow exponential growth. It starts at \( y = 2 \) at \( x = 0 \).

5. **\( f(x) = 2 \cdot 2^x \)**: This function has a base of 2, leading to faster exponential growth than the previous growth functions. It starts at \( y = 2 \) at \( x = 0 \) and grows rapidly for positive \( x \).

### Matching with the Graphs:
Using the general shape of each function, here is the likely matching of functions to graphs:
- **Graph with rapid growth** (likely the steepest line): \( f(x) = 2 \cdot 2^x \).
- **Graph with moderate growth**: \( f(x) = 2^x \).
- **Graph with slower growth**: \( f(x) = 2 \cdot 1.25^x \).
- **Graph with slow decay**: \( f(x) = 2 \cdot 0.8^x \).
- **Graph with rapid decay** (steepest descent): \( f(x) = 2 \cdot 0.5^x \).

### Explanation for Each Match:
- **Growth rate** is determined by the base. Bases above 1 indicate growth, while bases between 0 and 1 indicate decay.
- **Starting value** is \( c \) (in this case, 2 for all functions), setting each graph at \( y = 2 \) when \( x = 0 \).
  
Would you like a deeper explanation of each matching or further assistance with part b? 

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### Additional Practice Questions
1. How would the graph change if the coefficient \( c \) were increased for each function?
2. What is the effect of a negative exponent on each of these functions?
3. How would the graph of \( f(x) = 2 \cdot (-0.8)^x \) look compared to \( f(x) = 2 \cdot 0.8^x \)?
4. Why does a base between 0 and 1 result in an exponential decay function?
5. What real-world scenarios could these types of exponential functions represent?

### Tip
Remember that the base \( a \) of an exponential function \( f(x) = c \cdot a^x \) determines whether the function is growing or decaying. Bases larger than 1 lead to growth, while bases between 0 and 1 cause decay.

Aufgabe 1: Ordnen Sie den Funktionsgleichungen auf dem Rand den passenden Graphen zu und begründen Sie.

This task involves matching exponential functions with corresponding graphs, analyzing properties of exponential growth and decay. Suitable for Grades 9-11, the solution explains how to identify each function's behavior based on its base value.

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