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.â€‹
(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 â‰¥ 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 â‰¥ 1.)
To shift the base graph down one unit, we subtract 1 outside the absolute value symbol,:
(Notice that the range is now y â‰¥ â€“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; itâ€™s still y â‰¥ 0. Notice also that subtracting 1 moved the graph to right. Many students will mistakenly move the graph to the left because thatâ€™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; itâ€™s still y â‰¥ 0. Notice also that adding 1 moved the graph to left. Many students will mistakenly move the graph to the right because thatâ€™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(x â€“ c). To summarize, we have
To shift up c units: |
y = f(x) + c |
To shift down c units: |
y = f(x) â€“ c |
To shift to the right c units: |
y = f(x â€“ c) |
To shift to the left c units: |
y = f(x + c) |