Question

the function $f(x) = 2 \cdot 5^x$ can be used to represent the curve through the points $(1, 10)$, $(2, 50)$, and $(3, 250)$. what is the multiplicative rate of change of the function?
32
2
5
10

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( f(x)=a\cdot b^{x} \), where \( a \) is the initial value and \( b \) is the multiplicative rate of change.

Step2: Identify \( b \) in given function

The given function is \( f(x) = 2\cdot5^{x} \). Comparing with \( f(x)=a\cdot b^{x} \), we see that \( b = 5 \). We can also verify using the points: for \( x = 1 \), \( f(1)=2\cdot5^{1}=10 \); \( x = 2 \), \( f(2)=2\cdot5^{2}=50 \); \( x = 3 \), \( f(3)=2\cdot5^{3}=250 \). The ratio between consecutive outputs: \( \frac{50}{10}=5 \), \( \frac{250}{50}=5 \), so the multiplicative rate of change is 5.

Answer:

5