Which equation represents the function $f(x)=(1.6)^x$ after it has been translated 5 units up and 9 units to the right?
A. $g(x)=(1.6)^{x+5}-9$
B. $g(x)=(1.6)^{x+5}+9$
C. $g(x)=(1.6)^{x-9}+5$
D. $g(x)=(1.6)^{x+9}+5$
Answer (1)
To shift the function f ( x ) 9 units to the right, replace x with x − 9 , resulting in ( 1.6 ) x − 9 .
To shift the function 5 units up, add 5 to the function, resulting in ( 1.6 ) x − 9 + 5 .
The translated function is therefore g ( x ) = ( 1.6 ) x − 9 + 5 .
The final answer is g ( x ) = ( 1.6 ) x − 9 + 5 .
Explanation
Understanding the Problem We are given the function f ( x ) = ( 1.6 ) x and asked to find the equation of the function g ( x ) that results from translating f ( x ) 5 units up and 9 units to the right.
Horizontal Translation To translate a function f ( x ) horizontally by h units, we replace x with ( x − h ) . A translation to the right corresponds to a positive h , and a translation to the left corresponds to a negative h . In our case, we want to translate 9 units to the right, so we replace x with ( x − 9 ) . This gives us f ( x − 9 ) = ( 1.6 ) x − 9 .
Vertical Translation To translate a function f ( x ) vertically by k units, we add k to the function. A translation upwards corresponds to a positive k , and a translation downwards corresponds to a negative k . In our case, we want to translate 5 units up, so we add 5 to the function. This gives us f ( x − 9 ) + 5 = ( 1.6 ) x − 9 + 5 .
Final Equation Therefore, the equation of the translated function is g ( x ) = ( 1.6 ) x − 9 + 5 .
Examples
Imagine you are tracking the growth of a plant, and its initial height is modeled by the function f ( x ) = ( 1.6 ) x , where x represents time in days. If you start measuring the plant's height 9 days later and from a point 5 units higher due to a raised platform, the new height function g ( x ) = ( 1.6 ) x − 9 + 5 would represent the plant's height relative to your new starting point. This type of transformation is useful in various scenarios, such as adjusting data for different starting points or reference levels.
