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The function shown is \( f(x) = \left( \frac{1}{2} \right)^x - 6 \), which is an exponential decay function due to the base \( \frac{1}{2} \), a number between 0 and 1. Here’s how we analyze the features of this function:

1. **Asymptote**: Since the function shifts downward by 6, there is a horizontal asymptote at \( y = -6 \).
   
2. **Range**: The range of \( f(x) \) is \( (-6, \infty) \) because the exponential function never reaches \( -6 \) but can get infinitely close, and it extends to positive infinity.

3. **Domain**: Exponential functions have a domain of all real numbers, so \( (-\infty, \infty) \).

4. **End Behavior**:
   - **Left Side** (\( x \to -\infty \)): As \( x \) becomes very negative, \( f(x) \) approaches \( \infty \).
   - **Right Side** (\( x \to \infty \)): As \( x \) becomes very positive, \( f(x) \) approaches \( -6 \).

So, the completed statements would be:

- "The function \( f(x) \) is an **exponential** function with a **horizontal** asymptote of **\( y = -6 \)**."
- "The range of the function is **\((-6, \infty)\)**, and it is **all real numbers** on its domain of **\((- \infty, \infty)\)**."
- "The end behavior on the **left side** is as **\( x \to -\infty\), \( f(x) \to \infty \)**, and the end behavior on the **right side** is as **\( x \to \infty\), \( f(x) \to -6\)**."

Would you like further details or have any questions?

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Here are some related questions to deepen your understanding:

1. How does the base \( \frac{1}{2} \) affect the behavior of an exponential function?
2. What happens to the asymptote if the function were \( f(x) = \left( \frac{1}{2} \right)^x + c \) for any constant \( c \)?
3. How does the range of the function change if the function is reflected across the x-axis?
4. What would be the end behavior if the base were greater than 1 (e.g., \( f(x) = 2^x - 6 \))?
5. How would you find the y-intercept of this function?

**Tip**: For exponential functions of the form \( a \cdot b^x + k \), \( k \) shifts the graph vertically and determines the horizontal asymptote.

What are the features of the function f(x) = (1/2)^x - 6 graphed below?

Explore the features of the exponential function f(x) = (1/2)^x - 6, including asymptotes, domain, range, and end behavior. Understand the impact of the base and vertical shift on the function's graph. Suitable for high school students in Grades 10-12.

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