Question
Without graphing, determine whether the function represents exponential growth or exponential decay. Then find the \( y \)-intercept. \( f(x)=4(0.64)^{x} \) Choose the correct answer below. The function represents exponential decay. The function represents exponential growth. The y-intercept is \( \square \). (Type an integer or a decimal.)
Ask by Mccoy Delgado. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The function represents exponential decay, and the y-intercept is 4.
Solution
The given function is:
\[
f(x) = 4 \times (0.64)^x
\]
**1. Determining Growth or Decay:**
- **Base Analysis:** The base of the exponential function is \( 0.64 \).
- **Growth vs. Decay:**
- If the base \( b > 1 \), the function represents **exponential growth**.
- If the base \( 0 < b < 1 \), the function represents **exponential decay**.
Since \( 0.64 < 1 \), the function represents **exponential decay**.
**2. Finding the \( y \)-Intercept:**
The \( y \)-intercept of a function occurs where \( x = 0 \).
\[
f(0) = 4 \times (0.64)^0 = 4 \times 1 = 4
\]
**Conclusion:**
- **Exponential Behavior:** The function represents **exponential decay**.
- **\( y \)-Intercept:** The \( y \)-intercept is **4**.
**Final Answer:**
- The function represents exponential decay.
- The y-intercept is \( 4 \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
The function \( f(x) = 4(0.64)^{x} \) represents exponential decay because the base of the exponent, \( 0.64 \), is less than 1. In exponential decay, as \( x \) increases, the function value decreases towards zero. To find the y-intercept, we evaluate the function at \( x = 0 \): \( f(0) = 4(0.64)^{0} = 4(1) = 4 \). Therefore, the y-intercept is \( 4 \).
