Describe the end behavior of the graph of the function f(x)=−5(4)^x−4

For ∞∞, type in the word infinity. For −∞, type in -infinity (a minus sign followed by the word infinity). Make sure that you type in the word infinity with a lower case i

As x→−∞→f(x)

As x→∞, f(x)→

For the given function \( f(x) = -5(4)^x - 4 \), we analyze the end behavior by considering the limits as \( x \) approaches \( -\infty \) and \( \infty \).

### As \( x \to -\infty \):
- When \( x \to -\infty \), the term \( (4)^x \to 0 \) because \( 4^x \) approaches 0 as \( x \) becomes more negative.
- Therefore, \( f(x) \approx -5(0) - 4 = -4 \).

Thus:
\[
\text{As } x \to -\infty, f(x) \to -4
\]

### As \( x \to \infty \):
- When \( x \to \infty \), the term \( (4)^x \to \infty \) because exponential functions grow rapidly for positive large \( x \).
- This means \( f(x) \to -5(\infty) - 4 = -\infty \).

Thus:
\[
\text{As } x \to \infty, f(x) \to -\infty
\]

### Summary:
- As \( x \to -\infty \), \( f(x) \to -4 \).
- As \( x \to \infty \), \( f(x) \to -\infty \).

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Do you want further details or have any questions?

Here are 5 related questions for deeper understanding:
1. What is the horizontal asymptote of \( f(x) \)?
2. How does changing the base of the exponential function affect end behavior?
3. How would the end behavior change if the coefficient of \( (4)^x \) were positive?
4. What happens to the graph if you shift \( f(x) \) vertically or horizontally?
5. Can you describe the behavior of the function for small values of \( x \)?

**Tip:** When analyzing end behavior, focus on how the exponential part dominates for large values of \( x \).

Describe the end behavior of the graph of the function f(x) = -5(4)^x - 4. As x→−∞, f(x)→? As x→∞, f(x)→?

Understand the end behavior of the exponential function f(x) = -5(4)^x - 4 as x approaches infinity and negative infinity. This solution explains key concepts such as exponential growth and limits, suitable for students in Grades 9-12.

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