If you are given the graph of $g$, how could you graph $f(x)=\log x+3$?
For each point on the graph of $g$,
Answer (1)
Recognize that f ( x ) = lo g x + 3 is a vertical shift of the graph of y = lo g x .
The graph of f ( x ) is obtained by shifting the graph of y = lo g x upwards by 3 units.
For each point ( x , y ) on the graph of g ( x ) = lo g x , the corresponding point on the graph of f ( x ) is ( x , y + 3 ) .
Therefore, for each point on the graph of g , we add 3 to the y-coordinate. The answer is add 3 to the y-coordinate.
Explanation
Understanding the Problem We are given the graph of a function g ( x ) and asked how to graph f ( x ) = lo g x + 3 . The key is to recognize that f ( x ) is a transformation of the basic logarithmic function.
Identifying the Transformation The function f ( x ) = lo g x + 3 represents a vertical shift of the basic logarithmic function y = lo g x . Specifically, it shifts the graph of y = lo g x upwards by 3 units.
Applying the Vertical Shift If we are given the graph of g ( x ) and we know that g ( x ) = lo g x , then to obtain the graph of f ( x ) = lo g x + 3 , we simply shift the graph of g ( x ) vertically upwards by 3 units. This means that for each point ( x , y ) on the graph of g ( x ) , the corresponding point on the graph of f ( x ) will be ( x , y + 3 ) .
Conclusion Therefore, for each point on the graph of g , we add 3 to the y-coordinate.
Examples
Logarithmic functions are used to model many real-world phenomena, such as the Richter scale for earthquake magnitudes. If we have a graph showing the magnitude of earthquakes over time, adding a constant to the logarithmic scale would be equivalent to adjusting the baseline for measuring the magnitudes. For example, if we redefine the scale to start at a higher baseline, we would effectively be adding a constant to the logarithmic function.
