From Mac Guides
Grapher is a graphing calculator application included with Mac OS X 10.4 (Tiger) and later. Originally named Curvus Pro X, it was purchased by Apple from Arizona Software in 2004. Grapher is able to draw both 2D and 3D graphs.
Using Grapher
is the "equation pop-up" and is located on the far right of the equation field. To show the equation palette:
- Window → Show Equation Palette, or
→ Show Equation Palette, or
- shift-⌘-E
Some commonly used items (eg π, √ ) can be entered with a single keystroke when combined with the "option" modifier key. To view these special characters, open the keyboard viewer and hold down "option" or "option+shift".
Values & Definitions
Every Grapher expression has a value type: scalar, multi-valued, a range, or a 2D or 3D vector.
Types
| Type | Syntax Example | Notes
|
| scalar
| 1.5 | numeric constant
|
| π
| Built-in constant. You can type out the constant name ("pi"), and Grapher will replace it with the symbol ("π")
|
| multi-valued | {0, π, 2π}
| When used in an expression, the expression is evaluated for each value. The expression is then multi-valued. For example, {0,1,2} * π is equivalent to {0,π,2π}. For another example, with a:={0,π/2,π}, we have cos(a)={cos(0),cos(π/2),cos(π)}={1,0,-1}
|
| range | 0..1
| The number of dots, as long as there are at least 2, does not seem to matter.
|
| vector
|
| Use equation palette → Operators to enter vectors. A vector whose elements are the axes variables (e.g. ) is called an "axis vector".
|
Some operators and functions can take more than 1 type for an argument. For example, you can add a range and a scalar. The types of the arguments determines the type of the expression.
Equations in Grapher are either definitions or graph equations. Definitions let you define your own constants and expressions. To define a symbol, create a new equation, then enter the symbol name (and argument if it's a function), followed by "=" or ":=" and an expression. When defining a symbol, Grapher may misinterpret "=" and give syntax errors. You can also define a symbol inline at the end of a graph equation by appending a comma followed by the definition. Example definitions:
- scalar constant:
φ := (1+√5)/2
- function:
f(t) := t sin(t)
- inline:
y=mx+b,m=2,b=-1
Operators
Grapher has the standard arithmetic operators you find on calculators: ^ (power), *, /, + and -/. In addition are operators found on graphing calculators.
Operators
| Category | Priority | Operators | Input | Name | Notes
|
| Boolean
| 1
| = | = | equal
| Comparison operators.
|
| ≠ | !=, <> | not equal
|
| < | < | less than
|
| > | > | greater than
|
| ≤ | <= | less than or equal
|
| ≥ | >= | greater than or equal
|
| 2 | ! ¬ | ! equation palette → Symbols → ¬ | not
| Boolean negation.
|
| 3 | & ∧ | & equation palette → Symbols → ∧ | and
| Conjunction. "a & b" is true if both "a" and "b" are true. You can also enter "∧" by turning off shortcuts in the equation palette and typing "^".
|
| 4
| | ∨ | | equation palette → Symbols → ∨ | or
| Disjunction. "a | b" is true if at least one of "a" or "b" are true.
|
| ⊗ | equation palette → Symbols → ⊗ | xor, either-or, exclusive-or
| "a ⊗ b" is true if either "a" or "b" is true (not both).
|
| ∈ | equation palette → Symbols → ∈ | element of, is-in
| "x ∈ [-1,5]" is true if x is in the interval "[-1, 5]"
|
| ∉ | equation palette → Symbols → ∉ | not an element of, not-in
| "x ∉ [-1,5]" is true if x is not in the interval "[-1, 5]"
|
| Category | Priority | Operators | Input | Name | Notes
|
| Arithmetic
|
| √▯
sqrt(▯)
| option-v, equation palette → Standard → √▯
sqrt()
| square root
|
| ▯√▯
| equation palette → Standard → ▯√▯
| n-th root
|
| ▯▯
| ^
equation palette → Standard → ▯▯
| power
|
| Combinatorics
|
| equation palette → Operators →
| choose
| Number of ways of choosing k items from n items. nCk = n! / k!(n-k)!
|
| Interval
|
| 1 | ~ | ~ | complement
|
| 2 | * ∩ | * equation palette → Symbols → ∩ | intersection
|
| 3 | + ∪ | + equation palette → Symbols → ∪ | union
|
| − | − | difference
|
| Piecewise
|
|  | →  |
| Left column is for values, right column is for tests.
|
? : | ? : | if-then
| Value of "a?b" is "b" if "a" is true, undefined otherwise. "a:b" is equivalent to "a?b".
|
?: :: | ?: :: | if-then-else
| Value of "a?b:c" is "b" if "a" is true, "c" otherwise. "a:b:c" is equivalent to "a?b:c".
|
| Category | Priority | Operators | Input | Name | Notes
|
| Series
|
|
| option-s
→ ∑ equation palette → Operators → ∑
| Sum
|
| option-shift-P
→ ∏ equation palette → Operators → ∏
| Product
|
| Calculus | ∂ | option-d, → ∂ | partial differential
|
| ∂ / ∂ t equation palette → Operators →
| partial derivative wrt t
| Leibniz's notation. If you use the equation palette to enter the derivative operator, you can change the differential 't' in the normal way (selecting it & typing a replacement).
|
| ∂ ^ n / ∂ t ^ n equation palette → Operators →
| nth partial derivative wrt t
| You can change the differential 't' and degree 'n' in the normal way.
|
' ̇ | '
→  | derivative
| Derivative, using Lagrange's and Newton's notation, respectively. Repeat n times for the nth derivative.
|
| option-b
→ ∫ equation palette → Operators → ∫
| integral
| Performs numeric, rather than symbolic, integration
|
| Category | Priority | Operators | Input | Name | Notes
|
Parametric equations
Parametric equations are based on vectors and ranges. Rather than setting the default dependent variable "y" equal to an expression, you set an axis vector (eg
) equal to a vector expression. Independent variables can have any name but must have a range value. The independent variable can be defined inline or separately from the parametric equation. See examples below.
Examples
Screenshots
The 3D "Conchoid" example |
The 2D "Polar Logarithmic" example |
See also