The function $f(x)=\log x$ is transformed to produce $g(x)=\log (x)-3$. Identify the type of transformation and describe the change.
Answer (2)
The function g ( x ) is obtained by subtracting 3 from f ( x ) .
This represents a vertical translation.
The transformation is a vertical shift down by 3 units.
The final answer is a vertical translation down by 3 units: vertical translation down by 3 units .
Explanation
Understanding the Problem We are given two functions: f ( x ) = lo g x and g ( x ) = lo g x − 3 . Our goal is to identify the transformation that maps f ( x ) to g ( x ) and describe the change.
Identifying the Transformation Comparing the two functions, we see that g ( x ) is obtained by subtracting 3 from f ( x ) . This means that for any value of x , the value of g ( x ) is 3 units less than the value of f ( x ) .
Describing the Change This type of transformation is a vertical translation. Specifically, it's a vertical shift downwards. The graph of g ( x ) is the same as the graph of f ( x ) , but shifted 3 units down along the y-axis.
Conclusion Therefore, the transformation is a vertical translation (or shift) down by 3 units.
Examples
Imagine you are tracking the height of a plant over time using the function f ( x ) = lo g x , where x is the number of days since you planted the seed. Now, suppose you realize that your measuring tool was not properly calibrated and consistently showed a reading that was 3 units higher than the actual height. To correct for this error, you would subtract 3 from each measurement, resulting in the new function g ( x ) = lo g x − 3 . This transformation represents a vertical shift downwards by 3 units, effectively correcting the measurement error and giving you a more accurate representation of the plant's growth. Vertical translations are useful in various real-world scenarios, such as adjusting data sets, calibrating instruments, and modeling physical phenomena where a constant offset is present.
The function g ( x ) = lo g x − 3 is a result of a vertical translation of the original function f ( x ) = lo g x . This transformation shifts the graph of the logarithmic function downwards by 3 units. Hence, the transformation can be clearly defined as a vertical translation down by 3 units.
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