Graph of a function

Di/ck in the a/ss

Examples

Functions of one variable

 

Graph of the function f(x, y) = sin(x²) · cos(y²).

The graph of the function.

${\displaystyle f(x)={\begin{cases}a,&{\text{if }}x=1,\\d,&{\text{if }}x=2,\\c,&{\text{if }}x=3,\end{cases}}}$ 

is

${\displaystyle \{(1,a),(2,d),(3,c)\}.\,}$ 

The graph of the cubic polynomial on the real line

${\displaystyle f(x)=x^{3}-9x\,}$ 

is

${\displaystyle \{(x,x^{3}-9x):x{\text{ is a real number}}\}.\,}$ 

If this set is plotted on a Cartesian plane, the result is a curve (see figure).

Functions of two variables

 

Plot of the graph of f(x, y) = −(cos(x²) + cos(y²))², also showing its gradient projected on the bottom plane.

The graph of the trigonometric function

${\displaystyle f(x,y)=\sin(x^{2})\cos(y^{2})\,}$ 

is

${\displaystyle \{(x,y,\sin(x^{2})\cos(y^{2})):x{\text{ and }}y{\text{ are real numbers}}\}.}$ 

If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface (see figure).

Oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the function surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function:

${\displaystyle f(x,y)=-(\cos(x^{2})+\cos(y^{2}))^{2}\,}$ 

Normal to a graph

Given a function f of n variables: ${\displaystyle x_{1},\dotsc ,x_{n}}$ , the normal to the graph is

${\displaystyle (\nabla f,-1)}$ 

(up to multiplication by a constant). This is seen by considering the graph as a level set of the function ${\displaystyle g(x,z)=f(x)-z}$ , and using that ${\displaystyle \nabla g}$  is normal to the level sets.

Generalizations

The graph of a function is contained in a Cartesian product of sets. An X–Y plane is a cartesian product of two lines, called X and Y, while a cylinder is a cartesian product of a line and a circle, whose height, radius, and angle assign precise locations of the points. Fibre bundles are not Cartesian products, but appear to be up close. There is a corresponding notion of a graph on a fibre bundle called a section.

Tools for plotting function graphs

Hardware

• Graphing calculator
• Oscilloscope

Software

See List of graphing software

See also

• Asymptote
• Chart
• Concave function
• Convex function
• Contour plot
• Critical point
• Derivative
• Epigraph
• Slope
• Stationary point
• Tetraview
• Vertical translation
• y-intercept

References

External links

 

Wikimedia Commons has media related to Function plots.

• Graph of function, derivative and antiderivative plotter
• Weisstein, Eric W. "Function Graph." From MathWorld—A Wolfram Web Resource.