Example Functions and Graphs
On this page you will find examples of interesting functions, along with their graphs created with the plotter tools on mathonthecloud.com!This page will be updated regularly! Visit our social media pages on Facebook and Twitter/X to see the newest examples!
Click on a link below to only see examples of a certain function type.
Show all examples
Explicit function examples
Parametric function examples
Polar function examples
Implicit function examples
Displaying 25 examples of the type: implicit
Implicit Function Family: `a/2*x y (x^2-y^2)-x^2-y^2=0`
The Maltese Cross is an ancient symbol that dates back to 16th century Europe. It represents bravery and its imagery often serves as service medals all over the world. It can be represented by implicit and polar equations! Here, the parameter 'a' controls the size of the cross.
Go to the plotter page for this graph.⧉
tags: implicit polar
Implicit Function Family: `a^4 y^2+4 (x^2+y^2)^3-4 a^2 (x^2+y^2)^2`
🌼🌼The Dürer Folium is a rose curve with the polar functions `r=a*sin(θ/2) or r=a*cos(θ/2)`. (Though unequal, both of these equations will result in the same graph!) It also has a more complicated implicit equation!
Go to the plotter page for this graph.⧉
tags: polar implicit
Implicit Function Family: `x^4 (a^2-x^2)-(0.8*a)^4 y^2=0`
🏋️ This Dumbbell Curve is a sextic(6th-degree) curve with the equation `x^4 (a^2-x^2)-(0.8*a)^4 y^2=0`. Here, the parameter 'a' controls the overall size and the 'weight' ends of the dumbbell.
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function Family: `y^4-(a/3)^2 y^2-x^4+(a/2)^2 x^2=0`
The Devil's Curve is so named for the central lemniscate (figure-8) shape that resembles a diabolo. Of course, there's nothing evil about a math equation! 😈😈😈 Here it is graphed with a single changing parameter.
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function Family: `(x^2+y^2)^3-4*x^2*y^2*(a^2+1)=a/10000`
A family of implicit functions that graphs into clovers 🍀🍀! in `(x^2+y^2)^3-4*x^2*y^2*(a^2+1)=a/10000`, 'a' parameter determines the size of each clover.
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function Family: `y^2-x*(x^4-a)=0`
The Burnside Curve has the implicit function `y^2-x*(x^4-a)=0` and it has an interesting property when graphed! when the parameter 'a' is greater than 0, it separates into 2 distinct curves as shown below.
Go to the plotter page for this graph.⧉
tags: implicit
Multiple Function Types: A Bicycle Wheel
An offroad bike wheel represented with 3 functions! In the multi-type plotter, four types of functions are graphed simultaneously! This is useful when you want to visualize relationships and interactions between different types of equations.
Go to the plotter page for this graph.⧉
tags: explicit implicit polar parametric
Multiple Function Types: 3 Circles
Three ways to graph a circle! Standard form: `(x-a)^2+(y-b)^2=r^2`; Parametric form: `x=r*cos(t)+a, y=r*sin(t)+b`. 'a' and 'b' are x and y displacements, 'r' is the radius. In polar form, it is simply `r=radius` for a circle centered on the origin!
Go to the plotter page for this graph.⧉
tags: implicit polar parametric
Multiple Implicit Equations: Yin & Yang
Four implicit functions are needed to represent the yin yang symbol known as the taijitu. In Chinese philosophy, yin and yang symbolizes two opposite but interconnected, perpetuating forces that exist in all of nature. ☯️☯️☯️
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function Family: `abs(x/a)^r+abs(y/b)^r-1=0`
A superellipse ♦️ has the equation of `abs(x/a)^r+abs(y/b)^r-1=0` and the shape of the graph depends on the variables a, b, and r. a and b determines the horizontal and vertical size. Variable r determines how 'bloated' the diamond is, with r=1 giving the shape straight edges, When r=2, it becomes a circle!
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function Family: `x^2 y^2-a^2 (x^2+y^2)=0`
A Cruciform, or a cross, can be represented mathematically by the equation `x^2 y^2-a^2 (x^2+y^2)=0`. The variable 'a' determines how big the cross is when it's graphed. It also has the parametric form of `x=a*sec(t), y=a*csc(t)`
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function Family: `(x^2+y^2)^2-2a^2*(x^2-y^2)=0`
The symbol for infinity ∞ is known as a lemniscate in geometry. The lemniscate of Bernoulli has the implicit equation `(x^2+y^2)^2-2a^2*(x^2-y^2)=0` where 'a' are the foci points on the x-axis. It can also be represented as the Polar equation `r=a*cos(2*θ)^(1/2)` within a specified range for θ.
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function Family: `(x-1)(x^2+y^2)-a*x^2=0`
Conchoid of de Sluze is a family of curves with the implicit equation `(x-1)(x^2+y^2)-a*x^2=0`. It also has the polar equation `r=sec(theta)+a*cos(theta)`. They have very interesting shapes when graphed!
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function: `tan(x^2+y^2)=0`
The implicit equation `tan(x^2+y^2)=0` gives us a graph of concentric circles.
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function: `x*sin(x^2+y^2)+y=0`
The implicit equation `x*sin(x^2+y^2)+y=0` produces this interesting graph!
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function Family: `y^2/2+cos(x^2+y^2)-a/2=0, a=[1,2,3...15]`
A psychedelic 8-ball. From the implicit equations `y^2/2+cos(x^2+y^2)-a/2=0, a=[1,2,3...15]`
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function Family: `(x^2+y^2)^3-4x^2y^2*(a^2+1)-a/1000=0, a=[1,2,3...10]`
The equations `(x^2+y^2)^3-4x^2y^2*(a^2+1)-a/1000=0, a=[1,2,3...10]` produce clover-shaped curves. 🍀
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function Family: `x^2+y^2*a-a^2=0, a=[-9,-8,-7...9]`
An implicit function family with an independent parameter. `x^2+y^2*a-a^2=0, a=[-9,-8,-7...9]`
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function: `cos(20*(arctan((x-1)/y)+arctan2(y,x+1)))=0`
This implicit equation, `cos(20*(arctan((x-1)/y)+arctan2(y,x+1)))=0`, does a good job of representing magnetic field lines between two poles when graphed! 🧲
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function: `abs(sin(x^2+2*x*y))-sin(x-2*y)=0`
This is the graph of the equation `abs(sin(x^2+2*x*y))-sin(x-2*y)=0`
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function: `x^2+(y-(x^2)^(1/3))^2-a^2=0`
Here is an implicit function, `x^2+(y-(x^2)^(1/3))^2-a^2=0`, forming hearts when graphed! ❤️
Go to the plotter page for this graph.⧉
tags: implicit
Cassini oval: `(x^2+y^2)^2-2*(x^2-y^2)+1-a^4=0`
Implicit functions can look very interesting when graphed. This is a Cassini oval, a curve with 2 distinct foci, with the equation `(x^2+y^2)^2-2*(x^2-y^2)+1-a^4=0`
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function: `x^2+y^2-49=0`
Here is a graph visualized using the random testing, or Monte Carlo method. This is the implicit equation `x^2+y^2-49=0`, which results in a circle.
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function Family: `y^2-x*y+a*x^2-25=0`
When graphed, the equation `y^2-x*y+x^2-25=0` gives a tilted, elliptical shape. Observe how the graph changes when a parameter is introduced to the equation, `y^2-x*y+a*x^2-25=0`, where 'a' is the set of integers 1 to 10. .
Go to the plotter page for this graph.⧉
tags: implicit
Implicit Function Circle: `x^2+y^2-r^2=0`
On the Implicit Function (Multi) plotter, you can graph multiple functions at once! `x^2+y^2-r^2=0` is shown here, where r is a set of parameters representing the radius of these circles.
Go to the plotter page for this graph.⧉
tags: implicit