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) = x^{2} and g(x) = (x - 2)^{2} are shown below and it is easily seen that the graph of (x - 2)^{2} is that of x^{2} shifted 2 units to the right.

f(x) = x^{2}

g(x) = (x - 2)^{2}

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.

f(x) = √x

g(x) = √(x + 2)

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

f(x) = x^{2}

g(x) = x^{2} + 2

If c is negative, the graph is translated down as shown in the graph below.

f(x) = x^{2}

g(x) = x^{2} - 1

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)

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.

f(x) = |x|

g(x) = - |x|

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|