Question

question 7 (multiple choice worth 2 points) (exploring exponential functions mc) what is the percent rate of change for the function $y = 8 \cdot 0.4^x$? 40% 60% 20% 80%

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( y = a(1 - r)^x \) for decay, where \( r \) is the rate of decay (in decimal), or \( y = a(1 + r)^x \) for growth, where \( r \) is the rate of growth (in decimal). The given function is \( y = 8 \cdot 0.4^x \). We can rewrite \( 0.4 \) as \( 1 - r \), so we need to find \( r \) and then convert it to a percentage.

Step2: Find the decay rate

Set \( 1 - r = 0.4 \). Solving for \( r \), we subtract \( 0.4 \) from \( 1 \): \( r = 1 - 0.4 = 0.6 \).

Step3: Convert to percentage

To convert the decimal \( 0.6 \) to a percentage, we multiply by \( 100 \): \( 0.6 \times 100 = 60\% \). This means the function is decaying at a rate of \( 60\% \) (since it's a decay function, the percent rate of change is the decay rate expressed as a percentage).

Answer:

60% (the option corresponding to 60%)