# Syntax for Our Graphing Software

Our graphing software implemented by our graphing calculator and other graphers on this site, is designed to be intelligent, intuitive, and user-friendly. However, for those who want to use it to its full potential, it is helpful to understand the syntax and conventions for entering expressions. The following sections provide more information on the syntax that should be used and conventions for entering functions and equations, parametric equations, and point sets.

## Syntax for Functions

To graph a function, for example, f(x) = 3x2 + 2x + 1 type in 3x^2+2x+1

Or, when graphing in the polar coordinate system, if the expression is represented by r(θ) = 2 + 2θ + 1, type in 3θ^2+2θ+1

To type θ type ..t (two dots followed by t). You can also use x for θ. All x's are internally replaced by θ when graphing functions in polar coordinate system.

A is a graphing software that plots the graph of a function on a given domain in the polar coordinate system. The resulting graph is referred to as the polar graph or the polar curve of the function.

## Syntax for Equations

An is a more versatile tool than a function grapher, as it can graph any equation of the form g(x,y) = f(x,y), where each side of the equation can contain both x and y.

To graph an equation, for example, x^3-xy+2y^2 = 5x+2y+5   just type in the equation (using the "=" sign).

## Syntax for Parametric Curves

A is a graphing software that draws the range of a function p(t) = [f1(t), ..., fn(t)] on a given domain in a coordinate system. Such a graph is called the graph of the parametric equations x1 = f1(t), ..., xn = fn(t) or the parametric curve represented by the function p(t).

To graph a parametric curve represented, for example, by a function p(t) = [x(t), y(t)] = [sin(t), cos(t)]   for -π < t < π or equivalently, by the equations x(t) = sin(t) y(t) = cos(t) < t < π type in [sin(t), cos(t)]   dom=(-pi, pi)

Using [ ] to enclose x(t), y(t) is optional.

Or, when graphing in the polar coordinate system, if the expression is represented by p(t) = [r(t), θ(t)] = [sin(t), cos(t)]   for < t < π or equivalently, by the equations r(t) = sin(t) θ(t) = cos(t) < t < π type in [sin(t), cos(t)]   dom=(-pi, pi)

Using [ ] to enclose r(t), θ(t) is optional.

## Syntax for Points

A (or coordinates plotter) is a graphing software that graphs a set of points in a coordinate system. The coordinates of the points are given by ordered pairs (an, bn) where (an, bn) can be either the Cartesian or polar coordinates of the points.

To graph a set of points {(xi, yi) : i=1,...,n} type in x1, y1; x2, y2; ... xn, yn

Or, when graphing the points {(ri, θi) : i=1,...,n} in the polar coordinate system, type in r1, θ1; r2, θ2; ... rn, θn

To make it easier to use our graphing software, we have implemented the following conventions:

Our graphing software can graph functions and parametric curves on a specified interval (domain) using the notation dom=(a, b). If you do not specify a domain, the graphing software will automatically append a suitable interval to the expression. You can then adjust the end-points as desired.

If no interval is specified, the graphing software will append dom = (-∞, ∞) or dom = (0, 2π) to function expressions, depending on whether it is graphing in the Cartesian or polar coordinate system, respectively. For parametric expressions, the graphing software will append dom = (0, 2π) for both Cartesian and polar graphing. You can change the endpoints as desired.

In polar or parametric graphing, the specified intervals must be bounded. If they are not, any instances of will be replaced by constant values.

Our graphing software recognizes x, θ, t, and y as variables and is programmed to work intelligently. It automatically detects the type of expression as you type. If you insert a comma, the relevant input panel will change to indicate that a parametric expression is being entered. If you delete the comma, the input panel will switch back to the function entering mode.

The Graphing software will then automatically replace:

• t's and θ's with x's when graphing functions using the Cartesian coordinate system.
• x's and t's with θ's when graphing functions using the polar coordinate system.
• x's and θ's with t's when graphing parametric equations in both coordinate systems.

Remark: Although the variables x, θ, and t are intended to be used for functions (Cartesian/polar) and parametric curves, respectively, you can still use them interchangeably. For example, the function expression `xsin(t)` will be internally replaced by either `xsin(x)` or `θsin(θ)`, depending on the selected coordinate system. Similarly, the parametric expression `[xsin(t), θcos(x)]` will be replaced by `[tsin(t), tcos(t)]`. Clicking anywhere on the page will replace the variables in the input box with the appropriate ones.

Note: Unless the variable y is used in an equation, the graphing software will treat it as either an x or a θ, depending on the selected coordinate system. In parametric expressions, any instances of the variable y will be replaced with a t.