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!
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Explicit function examples
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Polar function examples
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Displaying 17 examples of the type: polar
Polar Function: `r=((1+sqrt(5))/2)^((2theta)/(pi))`
🌀🌀 The Golden Spiral is a logarithmic spiral with the growth factor of φ(phi), the golden ratio, which is approximately 1.618. Logarithmic spirals can often be found in nature, such as the shells of mollusks, in the form of hurricanes, and the shape of the Milky Way galaxy.🌀🌀
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tags: polar
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.
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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!
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tags: polar 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.
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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!
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tags: implicit polar parametric
Polar Function: `r=theta*cos(theta)`
The so-called 'Garfield Curve' is the polar equation `r=theta*cos(theta)`. When graphed from -2π to 2π, It somewhat represents the face of the famous cartoon cat Garfield. When range of a is extended, the graph becomes far more interesting.
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tags: polar
Polar Function: `r=sin(2*theta)`
A quadrifolium is a rose curve that is reminiscent of a four-leaf clover! It can be represented with an implicit equation, `(x^2+y^2)^3-4*x^2*y^2=0`, and also with a much simpler polar equation, `r=sin(2*theta)`.
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tags: polar
Polar Function: `r=3.5-1.5*abs(cos(theta))*abs((1.3+abs(sin(theta)))^(0.5))+cos(2theta)-3*sin(theta)+0.7*cos(12.2theta)`
A 'braided' polar curve that forms a heart! ❤️ `r=3.5-1.5*abs(cos(theta))*abs((1.3+abs(sin(theta)))^(0.5))+cos(2theta)-3*sin(theta)+0.7*cos(12.2theta)`
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tags: polar
Polar Function: `r=theta-30+sin(theta^2)`
A spiral graph that gains frequency as it tightens. `r=theta-30+sin(theta^2)`
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tags: polar
Polar Function: `r=(1+0.9*cos(8theta))(1+0.1cos(24theta))(0.9+0.1*cos(200theta))(1+sin(theta))`
A certain kind of plant leaf! This rather complex equation, `r=(1+0.9cos(8theta))(1+0.1cos(24theta))(0.9+0.1*cos(200theta))(1+sin(theta))`, produces a five-bladed leaf that is quite pleasant to the eyes.
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tags: polar
Polar Function: `r=theta+2*sin(2*pi*theta)+4*cos(2*pi*theta)`
This polar equation, `r=theta+2*sin(2*pi*theta)+4*cos(2*pi*theta)`, resembles the spiral shape of a camellia flower.
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tags: polar
Polar Function: `r=4*round(sin(1.2*theta))^2-round(cos(1.2*theta))^2+4*cos(1.2*theta)+sin(1.2*theta)`
The polar equation `r=4*round(sin(1.2*theta))^2-round(cos(1.2*theta))^2+4*cos(1.2*theta)+sin(1.2*theta)` produces a very complex curve!
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tags: polar
Polar Function: `r=10+sin(2*pi*theta)`
The equation `r=10+sin(2*pi*theta)` produces a braid-like, circular curve.
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tags: polar
Polar Function: `r=sin(8*theta/9)`
This is the polar equation `r=sin(8*theta/9)`. It's rather short, but produces a satisfying circular curve.
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tags: polar
Polar Function: `r=(sin(theta)*abs(cos(theta))^(1/2))/(sin(theta)+1.4)-2*sin(theta)+2`
You can create a cardioid (a heart shaped curve! ❤️) with the polar function `r=(sin(theta)*abs(cos(theta))^(1/2))/(sin(theta)+1.4)-2*sin(theta)+2`
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tags: polar
Polar Function: `r=a*cos(theta)*sin(theta)`
Here is the function `r=a*cos(theta)*sin(theta)`. By adding some complexity, the graph can take on interesting geometric formations!
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tags: polar
Polar Function: `r=cos(n*theta)`
In the Polar plotting tool, graphing `r=cos(n*theta)`, where n is an integer, can have interesting results! Here is the graph for `r=cos(9theta)`
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tags: polar