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Shift a Cosecant Function on a Graph

The cosecant function is affected by the same multiplication, addition, and subtraction principles that affect the other functions. For example, adding or subtracting a number to or from the cosecant function results in the graph going up or down. Adding or subtracting numbers to the angle variable slides the graph left or right.

The graph of <i>y</i> = csc (<i>x</i> + 2) + 2.
The graph of y = csc (x + 2) + 2.

The preceding figure shows two slides: moving the graph to the left by 2 units and up by 2 units. The equation of that graph is y = csc (x + 2) + 2.

Although the asymptotes are left out, you can still tell where they are — the shape of the graph is pretty clear.

The graph of <i>y</i> = 2csc 2<i>x</i>.
The graph of y = 2csc 2x.

Multiplying by a number changes the steepness and period of the cosecant function. If you multiply the function by 2, the curve gets steeper and has more space between its bottom and top. If you multiply the angle variable by 2, twice as much of the curve fits in the usual amount of horizontal space. The second figure shows both changes in the graph of y = 2csc 2x.

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