A step by step tutorial on the properties of transformations such as vertical and horizontal translation (or shift) , scaling and reflections on x-axis and y-axis of graphs of functions is presented.
Translation / Shifting Horizontally
Take any function f(x) and change x to x + c, the graph of f(x + c) will be the graph of f(x) shifted horizontally c units.
If c is negative, then the graph is shifted to the right. The example of the graph of f(x) = x2 and g(x) = (x - 2)2 are shown below and it is easily seen that the graph of (x - 2)2 is that of x2 shifted 2 units to the right.
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f(x) = x2 |
g(x) = (x - 2)2 |
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If c is positive, then the graph is shifted to the left. The example of the graph of f(x) = √(x) and g(x) = √(x + 2) are shown below and it is easily seen that the graph of √(x + 2) is that of √(x) shifted 2 units to the left.
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f(x) = √x |
g(x) = √(x + 2) |
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If we add a constant c to f(x), the graph of f(x) + c will be the graph of f(x) translated (or shifted) vertically. If c is positive, the graph is translated up as shown in the graph below.
Translation / Shifting Vertically
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f(x) = x2 |
g(x) = x2 + 2 |
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If c is negative, the graph is translated down as shown in the graph below.
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f(x) = x2 |
g(x) = x2 - 1 |
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Reflection on y-axis
Take any function f(x) and change x to - x, the graph of f(- x) will be the graph of f(x) reflected on the y axis.|
f(x) = √(x) |
g(x) = √(- x) |
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Reflection on x-axis
Take any function f(x) and it to - f(x), the graph of - f(x) will be the graph of f(x) reflected on the x axis.
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f(x) = |x| |
g(x) = - |x| |
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Scaling Vertically
Take any function f(x), the graph of k f(x) (with k > 0) will be the graph of f(x) expanded vertically if k is greater than 1 and compressed vertically if k is less than 1.
Graph in blue is that of f(x) = |x| and the graph in red is that of g(x) = 2 |x|
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