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
Parametric function examples
Polar function examples
Implicit function examples
Displaying 46 examples
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
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.
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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.
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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.
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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.
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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.
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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
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. ☯️☯️☯️
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tags: implicit
Multiple Explicit Functions: `y=a*sin(b*x+c)+d`
The different properties of the sine wave! ∿∿∿ In `y=a*sin(b*x+c)+d`, each of the variables change a property of the graph. a: amplitude, b: frequency, c: x-axis displacement. d: y-axis displacement.
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tags: explicit
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!
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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)`
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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.
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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!
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tags: implicit
Parametric Function: `x=cos(t)-(sin(t)^2)/(sqrt(2)), y=cos(t)*sin(t)`
The parametric function, `x=cos(t)-(sin(t)^2)/(sqrt(2)), y=cos(t)*sin(t)`, is known as the Fish Curve.
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tags: 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
Implicit Function: `tan(x^2+y^2)=0`
The implicit equation `tan(x^2+y^2)=0` gives us a graph of concentric circles.
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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!
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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]`
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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. 🍀
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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]`
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tags: implicit
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
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! 🧲
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tags: implicit
Parametric Function: `x=(1+tanh(10*sin(n*t))/10)*cos(t), y=(1+tanh(10*sin(n*t))/10)*sin(t)`
The parametric equations `x=(1+tanh(10*sin(n*t))/10)*cos(t), y=(1+tanh(10*sin(n*t))/10)*sin(t)` produces curves that are distinctly 'gear'-shaped, where 'n' is the number of teeth in the gear!
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tags: parametric
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
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`
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tags: implicit
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
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! ❤️
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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`
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tags: implicit
Parametric Function: `x=cos(cos(t))^2*(1+cos(1.92*t)^4), y=sin(sin(t))^2*sin(sin(1.92*t)^3)`
Parametric functions can produce fascinating images when they are graphed. Their complexity is only limited by the equations you create! Click this link to see an example!
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tags: parametric
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
Explicit Function Family: `y=(a*x)/(a^2+x^2), a={-5...5}`
This is the equation `y=(a*x)/(a^2+x^2), a={-5...5}`. Depending on the function, changes in the value of 'a' can alter the resulting graph dramatically!
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tags: explicit
Explicit Function Family: `y=a*sin(x)*x, a={-2...10}`
An explicit equation is quite simple. In y=f(x), y's value is dependent on x, the independent variable. `y=a*sin(x)*x` is shown here, where 'a' is a set of integers.
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tags: explicit
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.
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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. .
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tags: implicit
Parametric Function: The Butterfly Curve
The Butterfly Curve is a set of parametric equations, `x=sin(t)*(E^(cos(t))-2cos(4t)-sin(t/12)^5); y=cos(t)*(E^(cos(t))-2cos(4t)-sin(t/12)^5)`, that looks like its namesake!
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tags: parametric
Parametric function: `x=t*cos(t); y=t*sin(t)`
Parametric functions can be used to produce many interesting graphs. A typical Archimedean spiral can be produced with the equations `x=t*cos(t); y=t*sin(t)`
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tags: parametric
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.
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tags: implicit