Does the function $f(x)=\frac{1}{2} \cdot 3^x$ represent growth or decay? What is the y-intercept of $f(x)$?
Growth; $\left(0, \frac{1}{2}\right)$
Growth; $(0,3)$
Decay; $\left(0, \frac{1}{2}\right)$
Decay; $(0,3)$
Answer (1)
The function represents growth because the base of the exponential term (3) is greater than 1.
To find the y-intercept, evaluate the function at x = 0 .
Calculate f ( 0 ) = 2 1 \t 3 0 = 2 1 \t 1 = 2 1 .
The y-intercept is ( 0 , 2 1 ) .
Explanation
Analyzing the Problem We are given the function f ( x ) = 2 1 \t 3 x . We need to determine if this function represents exponential growth or decay and find its y-intercept.
Determining Growth or Decay To determine if the function represents growth or decay, we look at the base of the exponential term. If the base is greater than 1, the function represents growth. If the base is between 0 and 1, the function represents decay. In this case, the base is 3, which is greater than 1. Therefore, the function represents growth.
Finding the y-intercept To find the y-intercept, we need to find the value of the function when x = 0 . We substitute x = 0 into the function: f ( 0 ) = 2 1 \t 3 0
Calculating the y-intercept Since any number raised to the power of 0 is 1, we have 3 0 = 1 . Therefore, f ( 0 ) = 2 1 \t 1 = 2 1 The y-intercept is the point where the graph of the function intersects the y-axis, which occurs at x = 0 . Thus, the y-intercept is ( 0 , 2 1 ) .
Final Answer The function f ( x ) = 2 1 \t 3 x represents growth, and its y-intercept is ( 0 , 2 1 ) .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population of bacteria doubles every hour, the population can be modeled by an exponential function. Similarly, the amount of a radioactive substance decreases exponentially over time. Understanding exponential functions allows us to make predictions and analyze these phenomena.
