Mathematics » Functions II » Hyperbolic Functions

# Revision of Hyperbolic Functions

## Functions of the form $$y=\cfrac{a}{x}+q$$

Functions of the general form $$y = \cfrac{a}{x} + q$$ are called hyperbolic functions, where $$a$$ and $$q$$ are constants.

The effects of $$a$$ and $$q$$ on $$f(x) = \cfrac{a}{x} + q$$:

• The effect of $$q$$ on vertical shift

• For $$q>0$$, $$f(x)$$ is shifted vertically upwards by $$q$$ units.

• For $$q<0$$, $$f(x)$$ is shifted vertically downwards by $$q$$ units.

• The horizontal asymptote is the line $$y = q$$.

• The vertical asymptote is the $$y$$-axis, the line $$x = 0$$.

• The effect of $$a$$ on shape and quadrants

• For $$a>0$$, $$f(x)$$ lies in the first and third quadrants.

• For $$a > 1$$, $$f(x)$$ will be further away from both axes than $$y = \cfrac{1}{x}$$.

• For $$0<a<1$$, as $$a$$ tends to $$\text{0}$$, $$f(x)$$ moves closer to the axes than $$y = \cfrac{1}{x}$$.

• For $$a<0$$, $$f(x)$$ lies in the second and fourth quadrants.

• For $$a < -1$$, $$f(x)$$ will be further away from both axes than $$y = – \cfrac{1}{x}$$.

• For $$-1<a<0$$, as $$a$$ tends to $$\text{0}$$, $$f(x)$$ moves closer to the axes than $$y = -\cfrac{1}{x}$$.

 $$a<0$$ $$a>0$$ $$q>0$$ $$q=0$$ $$q<0$$

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