Which best describes the graph of $f(x)=\log _2(x+3)+2$ as a transformation of the graph of $g(x)=\log _2 x$?
A. a translation 3 units right and 2 units up
B. a translation 3 units left and 2 units up
C. a translation 3 units up and 2 units right
D. a translation 3 units up and 2 units left
Answer (1)
Replacing x with x + 3 in lo g 2 x translates the graph 3 units to the left.
Adding 2 to lo g 2 ( x + 3 ) translates the graph 2 units up.
Therefore, the graph of f ( x ) = lo g 2 ( x + 3 ) + 2 is obtained from the graph of g ( x ) = lo g 2 x by a translation 3 units left and 2 units up.
The answer is a translation 3 units left and 2 units up.
Explanation
Understanding the Problem We are given two functions: f ( x ) = lo g 2 ( x + 3 ) + 2 and g ( x ) = lo g 2 x . We want to describe how the graph of g ( x ) is transformed to obtain the graph of f ( x ) .
Identifying Transformations The function f ( x ) is obtained from g ( x ) by replacing x with x + 3 and adding 2. Replacing x with x + 3 corresponds to a horizontal translation. Adding 2 corresponds to a vertical translation.
Horizontal Translation Replacing x with x + 3 shifts the graph 3 units to the left. This is because if we want x + 3 to have the same value as x in the original function, x must be 3 units smaller.
Vertical Translation Adding 2 to the function shifts the graph 2 units up. This is because for any value of x , the value of the function is increased by 2.
Conclusion Combining these two transformations, we see that the graph of f ( x ) is obtained from the graph of g ( x ) by a translation 3 units left and 2 units up.
Examples
Understanding transformations of functions is crucial in many fields. For example, in signal processing, shifting a signal in time or frequency is a common operation. Similarly, in image processing, transformations like translations are used to align images or to track objects. The ability to recognize and apply these transformations allows engineers and scientists to manipulate data and extract meaningful information.
