nich graph represents an exponential function?

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 (\( b > 1 \)):**  *Graph of \( f(x) = 2^{x} \). Notice how the function rises steeply as \( x \) increases.* - **Exponential Decay Example (\( 0 < b < 1 \)):**  *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 \( b > 1 \). --- If you have specific graphs you’d like to analyze or further questions about exponential functions, feel free to share more details!