Hyperbolas, asymptotes and domain and range

The basic hyperbola

• A reciprocal or hyperbolic function has the variable in the denominator:

$$y = \frac{a}{x}$$

• Its graph has two separate branches that never meet, sitting in opposite quadrants.
• The sign of $a$ decides which pair:
  • $a > 0$ — branches in the first and third quadrants
  • $a < 0$ — branches in the second and fourth

Asymptotes

• An asymptote is a line the curve approaches ever more closely but never touches.

• For $y = \dfrac{a}{x}$:

  • Vertical asymptote at $x = 0$ — the function is undefined there, because you cannot divide by zero
  • Horizontal asymptote at $y = 0$ — as $x$ grows large, $\frac{a}{x}$ shrinks towards zero without reaching it

• Always give an asymptote as an equation, such as $x = 0$, not as "the $y$-axis" and never just as a dashed line.

The shifted hyperbola

$$y = \frac{a}{x - h} + k$$

• The asymptotes move with the graph:

  • Vertical asymptote: $x = h$ — found by setting the denominator to zero
  • Horizontal asymptote: $y = k$ — the constant added on the outside

• Watch the sign. $y = \dfrac{3}{x - 4} + 2$ has asymptotes at $x = 4$ and $y = 2$; $y = \dfrac{3}{x + 4}$ has its vertical asymptote at $x = -4$.

• To find the vertical asymptote of any rational function, set the denominator equal to zero and solve. That works whatever the numerator is.

Domain and range of a hyperbola

• Domain: all real $x$ except where the denominator vanishes.

  • For $y = \dfrac{a}{x-h} + k$, the domain is $x \neq h$.

• Range: all real $y$ except the horizontal asymptote.

  • The range is $y \neq k$.

• The excluded values are exactly the asymptotes. Once you have found the asymptotes, the domain and range follow immediately.

Domain and range across the families

• This is a named requirement of the standard — "properties of functions (may include domain and range)".

Function	Domain	Range
Linear $y = mx + c$ ($m \neq 0$)	all real $x$	all real $y$
Quadratic, $a > 0$, vertex $(h,k)$	all real $x$	$y \geq k$
Quadratic, $a < 0$	all real $x$	$y \leq k$
Cubic	all real $x$	all real $y$
Exponential $y = ab^x$ ($a>0$)	all real $x$	$y > 0$
Logarithm $y = \log_b x$	$x > 0$	all real $y$
Hyperbola $y = \frac{a}{x-h}+k$	$x \neq h$	$y \neq k$
$y = \sqrt{x}$	$x \geq 0$	$y \geq 0$

• The pattern to notice: a domain restriction always comes from something you cannot do — divide by zero, take the log of a non-positive number, or take the square root of a negative.

Restricted domains in context

• In a modelling problem, the domain is usually limited by the situation, not by the algebra.
• If $h = 20t - 5t^2$ models a ball's height, the model applies only while the ball is in the air — so the domain is $0 \leq t \leq 4$, even though the parabola itself is defined for all $t$.
• Stating the practical domain is a Merit-level skill, and it shows you understand what the model does and does not describe.

Is it a function?

• A relation is a function if each $x$-value gives exactly one $y$-value.
• The vertical line test: if any vertical line crosses the graph more than once, it is not a function.
• Circles fail; parabolas, cubics, exponentials, logarithms and hyperbolas all pass.