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# Transformations

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## Transformations

Now that we know how to extract basic information from a function and sketch its curve graphically, we can consider how to alter the function to translate, reflect or stretch the graph.

**Adding a constant to the function**

The graph of **f(x) + k** (where k is a constant), has the same shape of the function f(x) but is k units higher as the diagram shows:

**Therefore f(x) + k is a translation of f(x) by the vector **

**Adding a constant to the variable, x**

The graph of f(x + a) is also the same shape as f(x), but this time it moves the graph 'a' units to the left (i.e. the graph happens 'a' units earlier). **Taking the example above gives:**

**Therefore f(x + a) is a translation of f(x) by the vector **

Using both rules:

The graph similar to y = x^{2} that turns at the point (2, 5) would have equation y = (x − 2)^{2} + 5; and in general,

**f(x) = f(x − a) + b is a translation of f(x) by the vector**

**The graph of f(-x) is simply a reflection of f(x) in the y-axis.** This is because the values of x that were positive are now negative and the previous negative values of x are positive. This swap is the same as reflecting in the y-axis.

**For example:**

**The graph of - f(x) is a reflection of f(x) in the x-axis.** If this scenario the y values swap (what was positive becomes negative and vice versa).

**The graph of af(x) is a stretch scale factor a in the y-axis.** This is because all the y-values become 'a' times bigger.

The graph of f(ax) is also a stretch. This time the multiple affects the x-values. (Everything happens 'a' times quicker.) Therefore:

**The graph of f(ax) is a stretch scale factor 1/a in the x-axis.**

**Example:**

The graph of f(x) = 2 sin 2x is a stretch of sin x by scale factor 2 in the y-axis, and a stretch scale factor 1/a in the x-axis.

f(x) = sin x |
The original function... |

f(x) = 2 sin x | Stretches the original function in the y axis... |

f(x) = 2 sin 2x |
Compresses (the inverse of stretches) the 2 sin x function in the x-axis... |