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?

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.