Answer
An exponential function is a graph that shows a smooth curve rising or falling rapidly. It has a constant base \( b \) and follows the form \( f(x) = a \cdot b^{x} \). Key features include:
- **Rapid Growth or Decay:** The graph increases or decreases steeply as \( x \) increases.
- **Y-intercept at \( (0, a) \):** The graph crosses the y-axis at \( (0, a) \).
- **Horizontal Asymptote at \( y = 0 \):** The graph approaches but never touches the x-axis.
- **No X-intercept:** The graph never crosses the x-axis.
To identify an exponential function graph, look for a smooth, curved shape that consistently increases or decreases, crosses the y-axis at \( (0, a) \), and approaches the x-axis without touching it.
**Example:**
- \( f(x) = 2^{x} \) shows rapid growth.
- \( f(x) = \left(\frac{1}{2}\right)^{x} \) shows rapid decay.
**Real-World Examples:**
- Population growth with constant growth rate.
- Compound interest calculations.
- Radioactive decay processes.
**Comparison:**
- Unlike linear functions (straight lines) or polynomial functions (varied curves), exponential functions have a constant percentage growth or decay rate.
**Identification Tips:**
- Look for a smooth, curved graph that consistently increases or decreases.
- Check for a y-intercept at \( (0, a) \).
- Ensure the graph approaches the x-axis without crossing it.
If you have specific graphs to analyze or need further clarification, feel free to ask!
Solution
An **exponential function** is a mathematical function of the form:
\[ f(x) = a \cdot b^{x} \]
where:
- \( a \) is a constant (the initial value),
- \( b \) is the base of the exponential, and
- \( x \) is the exponent.
**Key Characteristics of Exponential Function Graphs:**
1. **Shape of the Graph:**
- **Increasing Exponential Functions (\( b > 1 \)):**
- The graph rises rapidly as \( x \) increases.
- Example: \( f(x) = 2^{x} \)
- **Decreasing Exponential Functions (\( 0 < b < 1 \)):**
- The graph decreases rapidly as \( x \) increases.
- Example: \( f(x) = \left(\frac{1}{2}\right)^{x} \)
2. **Domain and Range:**
- **Domain:** All real numbers (\( -\infty < x < \infty \))
- **Range:** Positive real numbers (\( 0 < f(x) < \infty \))
3. **Horizontal Asymptote:**
- The graph approaches but never touches the horizontal line \( y = 0 \) (the x-axis) as \( x \) approaches \( -\infty \).
4. **Intercepts:**
- **Y-intercept:** The graph always passes through the point \( (0, a) \) because \( b^{0} = 1 \), so \( f(0) = a \).
- **X-intercept:** Exponential functions do not have an x-intercept since \( f(x) \) never equals zero.
5. **Growth and Decay:**
- **Exponential Growth:** When \( b > 1 \), the function models phenomena that increase rapidly, such as population growth or compound interest.
- **Exponential Decay:** When \( 0 < b < 1 \), the function models phenomena that decrease rapidly, such as radioactive decay or depreciation.
6. **Smoothness and Continuity:**
- The graph is smooth and continuous for all real numbers without any breaks or gaps.
**Visual Identification:**
- **Exponential Growth Example (\( b > 1 \)):**
![Exponential Growth](https://i.imgur.com/EXpnEaM.png)
*Graph of \( f(x) = 2^{x} \). Notice how the function rises steeply as \( x \) increases.*
- **Exponential Decay Example (\( 0 < b < 1 \)):**
![Exponential Decay](https://i.imgur.com/Lmn5LwZ.png)
*Graph of \( f(x) = \left(\frac{1}{2}\right)^{x} \). Notice how the function decreases rapidly as \( x \) increases.*
**Comparison with Other Functions:**
- **Linear Functions (\( f(x) = mx + c \)):**
- Graph forms a straight line.
- Growth rate is constant.
- **Polynomial Functions (\( f(x) = x^{n} \)):**
- Graph shape varies based on the degree \( n \).
- Growth rate depends on the power.
- **Exponential Functions (\( f(x) = a \cdot b^{x} \)):**
- Graph is always curved, either increasing or decreasing.
- Growth rate is proportional to the current value, leading to rapid increases or decreases.
**Real-World Applications:**
- **Population Growth:** Modeling how populations increase over time with a constant growth rate.
- **Finance:** Calculating compound interest where the amount grows exponentially.
- **Physics and Chemistry:** Describing radioactive decay and half-life processes.
**How to Identify an Exponential Function Graph:**
When presented with multiple graphs, look for the following features to identify an exponential function:
1. **Curved Shape:** Unlike straight lines or polynomial curves, exponential graphs are smooth and either consistently increase or decrease.
2. **Asymptote:** The x-axis acts as a horizontal asymptote.
3. **Y-intercept:** The graph will cross the y-axis at \( (0, a) \).
4. **Rapid Change:** The function should exhibit rapid growth or decay, getting steeper as \( x \) moves away from zero in the direction dictated by the base \( b \).
**Example Problem:**
*Which of the following graphs represents an exponential growth function?*
1. A straight line passing through the origin with a positive slope.
2. A parabola opening upwards.
3. A curve that starts slow and increases rapidly, passing through \( (0, 1) \).
4. A horizontal line at \( y = 2 \).
**Solution:**
- **Option 3** is the correct choice. It describes a curve that starts slowly and then increases rapidly, characteristic of an exponential growth function like \( f(x) = b^{x} \) with \( b > 1 \).
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