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Translations of Graphs

Many graphs can be obtained by shifting a base graph around by adding positive or negative numbers to various places in the function.
 
Take for example, the absolute value function . Its graph is
 

 
(Notice that sometimes an arrow is added to a graph to indicate the graph continues indefinitely and sometimes nothing is used. To indicate that a graph stops, a dot is added to the terminal point of the graph. Also, notice that the domain of the absolute value function is all x because you can take the absolute value of any number. The range is y ≥ 0 because the graph touches the x-axis at the origin, is above the x-axis elsewhere, and increases indefinitely.)

To shift this base graph up one unit, we add 1 outside the absolute value symbol, :
 


(Notice that the range is now y ≥ 1.)

 

To shift the base graph down one unit, we subtract 1 outside the absolute value symbol,:



(Notice that the range is now y ≥ –1.)
 

To shift the base graph to the right one unit, we subtract 1 inside the absolute value symbol, :



 

(Notice that the range did not change; it’s still y ≥ 0. Notice also that subtracting 1 moved the graph to right. Many students will mistakenly move the graph to the left because that’s where the negative numbers are.)

To shift the base graph to the left one unit, we add 1 inside the absolute value symbol, :


(Notice that the range did not change; it’s still y ≥ 0. Notice also that adding 1 moved the graph to left. Many students will mistakenly move the graph to the right because that’s where the positive numbers are.)
 
The pattern of the translations above holds for all functions. So to move a function y = f(x) up c units, add the positive constant c to the exterior of the function: y = f(x) + c. To move a function y = f(x) to the right c units, subtract the constant c in interior of the function: y = f(xc). To summarize, we have
 

To shift up c units:

y = f(x) + c

To shift down c units:

y = f(x) – c

To shift to the right c units:

y = f(xc)

To shift to the left c units:

y = f(x + c)





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