Tanisha is graphing the function [tex]$f(x)=25\left(\frac{3}{5}\right)^x$[/tex]. She begins by plotting the point [tex]$(1,15)$[/tex]. Which could be the next point she plots on the graph?
[tex]$(2,9)$[/tex]
[tex]$(2,-10)$[/tex]
[tex]$\left(2,14 \frac{2}{5}\right)$[/tex]
[tex]$(2,5)$[/tex]
Answer (1)
Verify that the point ( 1 , 15 ) lies on the graph of the function f ( x ) = 25 ( 5 3 ) x .
Calculate f ( 2 ) = 25 ( 5 3 ) 2 = 9 .
Compare the calculated value of f ( 2 ) with the y-coordinates of the given options.
The correct point is ( 2 , 9 ) , so the answer is ( 2 , 9 ) .
Explanation
Verifying the Initial Point First, let's verify that the point ( 1 , 15 ) lies on the graph of the function f ( x ) = 25 ( 5 3 ) x . We can do this by substituting x = 1 into the function and checking if f ( 1 ) = 15 . f ( 1 ) = 25 ( 5 3 ) 1 = 25 × 5 3 = 5 × 3 = 15 Since f ( 1 ) = 15 , the point ( 1 , 15 ) does indeed lie on the graph of the function.
Calculating f(2) Next, we need to find the y-coordinate of the point on the graph where x = 2 . We can do this by substituting x = 2 into the function: f ( 2 ) = 25 ( 5 3 ) 2 = 25 × ( 5 3 ) × ( 5 3 ) = 25 × 25 9 = 9 So, the point ( 2 , 9 ) lies on the graph of the function.
Finding the Correct Point Now, we compare the calculated value of f ( 2 ) with the y-coordinates of the given options. The options are: ( 2 , 9 ) ( 2 , − 10 ) ( 2 , 14 5 2 ) ( 2 , 5 ) Since f ( 2 ) = 9 , the correct point is ( 2 , 9 ) .
Examples
Exponential functions like the one in this problem are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a city's population is decreasing at a rate of 40% per year, and the initial population is 25,000, the population after x years can be modeled by the function f ( x ) = 25000 ( 5 3 ) x . Understanding how to plot and analyze such functions is crucial for making predictions and informed decisions in these scenarios.
