Comparison of exponential growth and decay graphs, demonstrating how the base k affects the function y = k^x.

Problem Analysis:

The image provided illustrates the concept of exponential functions and their graphs. Specifically, it shows two types of exponential functions:
1. Exponential Growth (\( k > 1 \)): The function \( y = 2^x \) is an example where the base \( k = 2 \) is greater than 1.
2. Exponential Decay (\( k < 1 \)): The function \( y = \left(\frac{1}{2}\right)^x \) is an example where the base \( k = \frac{1}{2} \) is less than 1.

The task appears to be related to understanding and analyzing these exponential functions based on their behavior as \( x \) increases or decreases. Let us break this down step by step.

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Key Concepts from the Image:

1. General Form of Exponential Function:
\[
y = k^x
\]
- \( k \) is the base of the exponential function.
- \( x \) is the exponent.

2. Behavior of the Function:
- Exponential Growth (\( k > 1 \)): As \( x \) increases, \( y \) increases rapidly. The graph rises steeply.
- Exponential Decay (\( k < 1 \)): As \( x \) increases, \( y \) decreases. The graph approaches zero asymptotically.

3. \( y \)-Intercept:
- For any exponential function \( y = k^x \), the \( y \)-intercept occurs at \( x = 0 \).
- Since \( k^0 = 1 \) for any \( k \neq 0 \), the \( y \)-intercept is always \( (0, 1) \).

4. Examples Provided:
- Exponential Growth: \( y = 2^x \) (base \( k = 2 \)).
- Exponential Decay: \( y = \left(\frac{1}{2}\right)^x \) (base \( k = \frac{1}{2} \)).

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Solution Explanation:

#### 1. Exponential Growth (\( k > 1 \)):
- Function: \( y = 2^x \)
- Base: \( k = 2 \) (greater than 1).
- Behavior:
- As \( x \) increases, \( y \) increases exponentially.
- For example:
- \( x = 0 \): \( y = 2^0 = 1 \)
- \( x = 1 \): \( y = 2^1 = 2 \)
- \( x = 2 \): \( y = 2^2 = 4 \)
- \( x = 3 \): \( y = 2^3 = 8 \)
- The graph rises steeply as \( x \) increases.

#### 2. Exponential Decay (\( k < 1 \)):
- Function: \( y = \left(\frac{1}{2}\right)^x \)
- Base: \( k = \frac{1}{2} \) (less than 1).
- Behavior:
- As \( x \) increases, \( y \) decreases exponentially.
- For example:
- \( x = 0 \): \( y = \left(\frac{1}{2}\right)^0 = 1 \)
- \( x = 1 \): \( y = \left(\frac{1}{2}\right)^1 = \frac{1}{2} \)
- \( x = 2 \): \( y = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)
- \( x = 3 \): \( y = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \)
- The graph approaches zero as \( x \) increases.

#### 3. Common Characteristics:
- \( y \)-Intercept: Both functions have a \( y \)-intercept at \( (0, 1) \) because \( k^0 = 1 \) for any \( k \neq 0 \).
- Domain: The domain of both functions is all real numbers (\( x \in \mathbb{R} \)).
- Range:
- For exponential growth (\( k > 1 \)), the range is \( (0, \infty) \).
- For exponential decay (\( k < 1 \)), the range is also \( (0, \infty) \).

#### 4. Asymptotic Behavior:
- Exponential Growth: As \( x \to -\infty \), \( y \to 0 \). The graph approaches the \( x \)-axis asymptotically.
- Exponential Decay: As \( x \to \infty \), \( y \to 0 \). The graph approaches the \( x \)-axis asymptotically.

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Final Answer:

The image illustrates the fundamental properties of exponential functions:
- Exponential Growth (\( k > 1 \)): The function \( y = 2^x \) increases rapidly as \( x \) increases.
- Exponential Decay (\( k < 1 \)): The function \( y = \left(\frac{1}{2}\right)^x \) decreases as \( x \) increases and approaches zero asymptotically.

\[
\boxed{\text{Exponential Growth: } y = 2^x \text{ (increases as } x \text{ increases); Exponential Decay: } y = \left(\frac{1}{2}\right)^x \text{ (decreases as } x \text{ increases)}}
\]