The parent function $f(x)=1.5^x$ is translated such that the function $g(x)=1.5^{x+1}+2$ represents the new function. Which is the graph of $g(x)$?
Answer (2)
The function g ( x ) = 1. 5 x + 1 + 2 is a transformation of f ( x ) = 1. 5 x .
The graph shifts 1 unit to the left and 2 units upward.
The y-intercept of g ( x ) is 3.5 .
The horizontal asymptote of g ( x ) is y = 2 .
Explanation
Understanding the Transformation The problem describes a transformation of the parent exponential function f ( x ) = 1. 5 x into a new function g ( x ) = 1. 5 x + 1 + 2 . We need to determine the characteristics of the transformed function g ( x ) to identify its graph.
Analyzing Horizontal and Vertical Shifts The function g ( x ) is derived from f ( x ) through two transformations:
Horizontal Translation: The term ( x + 1 ) in the exponent indicates a horizontal shift. Specifically, it represents a shift of the graph 1 unit to the left.
Vertical Translation: The addition of + 2 to the entire expression indicates a vertical shift. Specifically, it represents a shift of the graph 2 units upward.
Finding the Y-Intercept To further understand the graph, let's find the y-intercept of g ( x ) . This is done by setting x = 0 :
g ( 0 ) = 1. 5 0 + 1 + 2 = 1. 5 1 + 2 = 1.5 + 2 = 3.5
So, the y-intercept is at the point ( 0 , 3.5 ) .
Determining the Horizontal Asymptote Next, let's determine the horizontal asymptote of g ( x ) . The parent function f ( x ) = 1. 5 x has a horizontal asymptote at y = 0 . Due to the vertical shift of 2 units upward, the horizontal asymptote of g ( x ) is y = 2 .
Conclusion In summary, the graph of g ( x ) is the graph of f ( x ) shifted 1 unit to the left and 2 units upward. It has a y-intercept at ( 0 , 3.5 ) and a horizontal asymptote at y = 2 .
Examples
Exponential functions and their transformations are used in various real-world scenarios, such as modeling population growth, radioactive decay, and compound interest. For example, if a population of bacteria grows at a rate of 50% per hour, the function f ( x ) = 1. 5 x can model the population size after x hours. If we introduce a new condition that shifts the starting time by 1 hour and adds a constant number of new bacteria, the transformed function g ( x ) = 1. 5 x + 1 + 2 can model the new population size. Understanding these transformations helps in predicting and analyzing real-world phenomena.
The function g ( x ) = 1. 5 x + 1 + 2 is a transformation of f ( x ) = 1. 5 x that shifts the graph 1 unit to the left and 2 units upward. It has a y-intercept at ( 0 , 3.5 ) and a horizontal asymptote at y = 2 .
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