Thank you for the clarification. The function you've identified, \( f(x) = \left(\frac{1}{5}\right)^x - 1 \), corresponds to an exponential function that has undergone a transformation.

### Key Characteristics of the Correct Function:
1. **Exponential Decay**: The base \( \frac{1}{5} \) is between 0 and 1, which causes the graph to decay as \( x \) increases.
2. **Vertical Shift**: The subtraction of 1 from the function \( \left(\frac{1}{5}\right)^x \) shifts the graph downward by 1 unit.
3. **Asymptote**: The horizontal asymptote of the function is at \( y = -1 \), as the function approaches but never reaches this value for large \( x \).

### Equation:
\[
f(x) = \left(\frac{1}{5}\right)^x - 1
\]

This means the graph represents an exponential decay function that has been shifted down by 1 unit.

Would you like any further details or explanations?

Here are some follow-up questions:
1. How do the base values of an exponential function (greater than 1 vs. between 0 and 1) affect the graph?
2. What is the effect of vertical shifts on the horizontal asymptote of an exponential function?
3. How does the function \( \left(\frac{1}{5}\right)^x \) differ from the function \( \left(\frac{5}{1}\right)^x \) in terms of graph behavior?
4. What are the general rules for graphing transformations such as shifts and reflections?
5. How would the graph change if the equation were \( f(x) = \left(\frac{1}{5}\right)^x + 1 \)?

**Tip:** When analyzing exponential functions, always check for any shifts that modify the asymptote, as these are key to identifying the correct equation.

Explore the function f(x) = (1/5)^x - 1, an exponential decay function shifted downward by 1 unit. Learn about exponential decay, vertical shifts, and asymptotes. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation