Which best describes the graph of [tex]f(x)=\log _2(x+3)+2[/tex] as a transformation of the graph of [tex]g(x)=\log _2 x[/tex]?
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)
The function f ( x ) = lo g 2 ( x + 3 ) + 2 is a transformation of g ( x ) = lo g 2 x .
Replacing x with x + 3 shifts the graph 3 units to the left.
Adding 2 to the function shifts the graph 2 units up.
The transformation is a translation 3 units left and 2 units up. 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 the transformation that maps the graph of g ( x ) to the graph of f ( x ) .
Analyzing Horizontal Shift First, let's analyze the horizontal transformation. In the function f ( x ) , we have lo g 2 ( x + 3 ) . This means that the argument of the logarithm is x + 3 instead of x . This corresponds to a horizontal shift. Since we are replacing x with x + 3 , the graph shifts 3 units to the left.
Analyzing Vertical Shift Next, let's analyze the vertical transformation. In the function f ( x ) , we have + 2 outside the logarithm. This means that the entire function is shifted upwards by 2 units.
Combining Transformations Combining these two transformations, we see that the graph of g ( x ) = lo g 2 x is translated 3 units to the left and 2 units up to obtain the graph of f ( x ) = lo g 2 ( x + 3 ) + 2 .
Final Answer Therefore, the correct answer is a translation 3 units left and 2 units up.
Examples
Understanding transformations of functions is crucial in various fields. For instance, 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 or manipulate images. Knowing how to describe these transformations mathematically allows engineers and scientists to precisely control and analyze these processes. For example, if you have a sound wave represented by g ( x ) = sin ( x ) , then f ( x ) = sin ( x − 2 ) + 3 would represent the same sound wave shifted 2 units to the right and 3 units up.
