Which graph models the function f(x) = −4(2)x? (2 points)

To model the function \( f(x) = -4 \cdot 2^x \), we need to understand its characteristics and how they translate to a graph.

1. **Understanding the Function:**
   - The base function is \( 2^x \), an exponential growth function.
   - The factor \(-4\) affects the amplitude and direction:
     - The coefficient \( -4 \) causes a vertical stretch by a factor of 4 and reflects the graph over the x-axis.

2. **Key Features of the Graph:**
   - **Y-Intercept:** At \( x = 0 \), \( f(0) = -4 \cdot 2^0 = -4 \).
   - **Behavior for Positive \( x \):** As \( x \) increases, \( 2^x \) increases, but because of the negative sign, \( f(x) \) decreases (becomes more negative).
   - **Behavior for Negative \( x \):** As \( x \) decreases, \( 2^x \) approaches 0, but because of the negative sign, \( f(x) \) approaches 0 from the negative side.
   - **Horizontal Asymptote:** The graph will have a horizontal asymptote at \( y = 0 \), but it will approach 0 from the negative side.

3. **Sketching the Graph:**
   - Start at the y-intercept (0, -4).
   - As \( x \) increases, the values become more negative.
   - As \( x \) decreases, the values approach 0 from the negative side.

Based on this understanding, you can look for a graph that matches these characteristics:

- Starts at (0, -4).
- Decreases rapidly as \( x \) increases.
- Approaches 0 from the negative side as \( x \) decreases.

Would you like more details or have any questions?

Here are 5 questions you might find interesting to ask next:
1. How do you determine the domain and range of an exponential function?
2. Can you explain how to transform exponential functions with different coefficients?
3. What are the key properties of logarithmic functions?
4. How do you solve exponential equations with different bases?
5. What is the significance of the horizontal asymptote in exponential functions?

**Tip:** When graphing functions, it’s helpful to plot key points and understand the overall shape and behavior of the graph, especially for exponential and logarithmic functions.

Explore how to graph the exponential function f(x) = -4 * 2^x with detailed steps and characteristics. Learn about its y-intercept, behavior for positive and negative x values, and the significance of the horizontal asymptote. Suitable for high school students studying exponential functions.

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