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# Types of functions

## Circle and ellipse

The distance $d$ between two points $(x,y)$ and $(x_0,y_0)$ is $$d\big((x,y),(x_0,y_0)\big) = \sqrt{(x-x_0)^2+(y-y_0)^2}.$$

The equation of a circle centred at $(x_0,y_0)$ with radius $r\gt 0$ is $$(x-x_0)^2+(y-y_0)^2 = r^2.$$ It consists of all the points that have a distance $r$ to the centre, since the above identity can be written as $d\big((x,y),(x_0,x_0)\big) = r$.

The equation of an ellipse centred at $(x_0,y_0)$ with radius $a$ and $b$ is $$\frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2} = 1$$

The ellipse passes through the points $(x_0\pm a,y_0)$ and $(x_0,y_0\pm b)$.

An ellipse is the result of dilating a circle of radius 1 by a ratio of $a$ along the $x$-axis and a ratio of $b$ along the $y$-axis.

Circle (A) and ellipse (B)

## Parametric functions

A parametric function expresses the coordinates of a point as a function of another variable called the parameter . This is used mainly in physics (mechanics) where the parameter is time.

In the plane, the usual notation is $$\big(x(t),y(t)\big)$$ for the function and $t$ for the parameter.

A function $f(x)$ is a parametric function with $x(t)=t$ and $y(t) = f(t)$.

The parametric equation for the circle is $$x(t) = \cos t,\quad y(t) = \sin t,\quad t\in\R.$$ This is seen immediately as $x^2(t)+y^2(t) = 1$ for all $t$.

## Rational functions

A rational function is the ratio of two polynomials $$f(x) = \frac{P(x)}{Q(x)}$$ It can always be in reduced form, i.e. with distinct zeros for $P$ and $Q$.

The degree $\operatorname{\textrm{deg}}(P)$ of a polynomial $P$ is the highest power of $P$.

The degree of $x+1$ is $1$ and the degree of $x^3-1$ is $3$.

The main properties of a rational function in reduced form are as follows.

• Domain. Everything but the zeros of $Q$.
• Vertical asymptote at each zero of $Q$.
• Zeros are the zeros of $P$.
• Horizontal and oblique asymptotes if $\operatorname{\textrm{deg}}(P) \le \operatorname{\textrm{deg}}(Q) + 1$. In this case, we write $f$ in the form $$ax + b + \frac{R(x)}{Q(x)}$$ where $\operatorname{\textrm{deg}}(R) \le\operatorname{\textrm{deg}}(Q)-1$. The asymptote is $y = ax+b$.

$$f(x) = \frac{x^3-x}{x^2-4}= \frac{x^3-4x+3x}{x^2-4} = x + \frac{ 3x}{x^2-4}$$ is a rational function in reduced form, because the zeros of the numerator ($-1$, $0$ and $1$) and denominator ($-2$ and $2$) are distinct. The function has asymptotes $x=-2$, $x=2$ and $y=x$.

## Hyperbola

The equation of a hyperbola centred at $(x_0,y_0)$, with transverse $x$-axis, and radii $a\gt0$ and $b\gt0$ is $$\frac{(x-x_0)^2}{a^2}-\frac{(y-y_0)^2}{b^2} = 1$$ It passes through $(x_0\pm a,y_0)$ and has asymptotes $$\displaystyle \frac{y}{b} = \pm\frac{x}{a}.$$ The two branches intersect the transverse $x$-axis.

We divide the equation by $x^2$ and send $x\to\infty$ to find the asymptotes $\displaystyle \lim_{x\to\infty}\frac{y^2}{x^2} = \frac{b^2}{a^2}.$

The equation of a hyperbola with transverse $y$-axis is $$\frac{(y-y_0)^2}{b^2} -\frac{(x-x_0)^2}{a^2}= 1.$$

Hyperbola with transverse $y$-axis (A) and hyperbola with transverse $x$-axis (B)

## Implicit functions

A curve can be defined implicitly by a relation of the form $$F\big(y(x),x\big) = 0.$$

For instance, a general function $y=f(x)$ can be rewritten as $y-f(x) = 0$.

A circle is defined implicitly by the relation $$y^2 + x^2 = 1.$$

The curve is not defined directly as a function of $x$. It must be deduced from the equation, and in general corresponds to the graph of several explicit functions.

This method of defining a curve is used for those where it would be too complex to write the equation explicitly.

For instance, the circle combines the graphs of two explicit functions $$y_+=\sqrt{1-x^2},\quad y_-=-\sqrt{1-x^2},\qquad x\in[-1,1].$$

Implicit and explicit equation of a circle

## Examples of rational functions

Examples of rational functions are $$f(x) = \frac{ax+b}{dx+e},\quad g(x) = \frac{ax^2+bx+c}{dx+e}.$$ We assume that $a,d\ne 0$ and that $-e/d$ is not a zero of the numerator, so that the functions are in reduced form

• Domain. The functions are defined on $\R\setminus \{-e/d\}$
• Zero. Their zeros are the zeros of the numerator.
• Vertical asymptote: $x=-e/d$.
• Horizontal asymptote for $f$: $\displaystyle y = a/d$.
• Oblique asymptote for $g$. $$\displaystyle y = \alpha x + \beta,\quad \alpha = \frac{a}{d},\; \beta=\frac{bd-ae}{d^2}.$$

The oblique asymptote for $g$ is found from the computations ($x\to\infty$) \begin{align*} g(x) - \alpha x-\beta &= \frac{ ax^2+bx+c -\alpha d x^2 - (\alpha e + \beta d)x - \beta e}{ dx+e }\\ &= \frac{cd^2-bde+ae^2}{d^2(dx+e)}\to 0. \end{align*}

Rational functions: $f(x)=\frac{x}{x+1}$ and $g(x)=\frac{x^2-1}{x}$