Question

describe the end behavior and the percent rate of change of the exponential function shown on the graph. as x increases, y increases, as x decreases, y approaches dropdown with options positive infinity, negative infinity, zero. the percent rate of change of this function is dropdown %.

Explanation:

Step1: Analyze end behavior as x decreases

For the exponential function \( y = 2^x \), when \( x \) decreases (approaches \( -\infty \)), we calculate the limit: \( \lim_{x \to -\infty} 2^x=\lim_{x \to -\infty} \frac{1}{2^{-x}}=\lim_{t \to \infty} \frac{1}{2^{t}} \) (let \( t=-x \)). As \( t \to \infty \), \( 2^{t} \to \infty \), so \( \frac{1}{2^{t}} \to 0 \). Thus, as \( x \) decreases, \( y \) approaches 0.

Step2: Determine percent rate of change

The general form of an exponential function is \( y = a(1 + r)^x \), where \( r \) is the percent rate of change. For \( y = 2^x \), we can write it as \( y = 1(1 + r)^x \), so \( (1 + r)=2 \), which gives \( r = 1 = 100\% \).

Answer:

As \( x \) decreases, \( y \) approaches zero; The percent rate of change of this function is \( 100\% \).