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Transformation Calculator Online, Step by Step, With points

An online transformation calculator provides an output function that has been translated into the Laplace form. Our online transformation calculator is simple and straightforward to use, with the result appearing in a matter of seconds.

However, the Laplace transform is named after Pierre Simon De Laplace, a prominent French mathematician (1749-1827). However, the Laplace transform, like all transforms, converts one signal into another using a set of rules or equations. Meanwhile, the application of the Laplace transformation is the best approach to turn differential equations into algebraic equations.

Transformation Calculator

However, in control system engineering, the Laplace transformation is extremely important. We may perform the laplace transforms of various functions in order to analyse the control system. Meanwhile, in order to analyse the dynamic control system, the properties of the Laplace transform and the inverse Laplace transformation are used. We will also go over the definition of the Laplace transform, its formula, properties, the Laplace transform table, and its applications in depth in this post.

Describe Transformation Calculator

In general, transformation is the process of converting an expression, a figure, or any function into another without changing its value. Translation, reflection, and rotation are also the most common transformations. When it comes to mathematical transformations, we can distinguish between two types: Z transformation and Laplace transformation. Each transformation will adhere to a set of guidelines. However, the integral transform of a given derivative function is the laplace transform.

Transformation Calculator with steps

However, the following is how to use the transforms calculator:

First, Step 1: In the input field, type any function you like.

Then, step 2: To acquire the result, click the “Submit” button.


Then, step 3: However, in the new window, the Laplace transform of the supplied function will be presented.

Transformation Calculator Definition

In most cases, we solve differential equations using Laplace transformations. However, the time-domain differential equation is first translated into an algebraic equation in the frequency-domain form. However, the algebraic equation is solved in the frequency domain first, then converted to the time domain to solve the differential equation. To put it another way, a Laplace transformation is just a quick way to solve a differential equation.

Read Also: Slant Asymptote Calculator Online,

There are other types of transformations, but the most well-known are Laplace and Fourier transforms. However, the Laplace transform is commonly used to reduce a complex differential equation to a straightforward algebraic issue. Even as the problem becomes more complicated, algebra is easier to solve than differential equations.

Transformation Calculator Formula

However, the integral transform of a given derivative function with real variable t into a complex function with variable s is known as the Laplace transform. Let f(t) be supplied for t 0, and assume that the function meets certain constraints that will be presented subsequently.

When the improper integral converges, the Laplace transform of f(t), denoted by Lf(t) or F(s), is determined by the Laplace transform formula:

Standard notation is as follows: We shall use an uppercase letter to represent the Laplace transform if the notation is apparent, e.g., L(f; s) = F. (s).

However, the one-sided Laplace transform is another name for the Laplace transform we defined. There is a two-sided variation with the integral changing from to.

Calculator for Transformations: Definition of Laplace Transformation

It is necessary to comprehend not only the tables but also the formula in order to grasp the Laplace transform.

To grasp the Laplace transform formula, consider the following: First Let f(t) be a function of t, with t larger than or equal to 0.

However, the Laplace transform of f(t), F(s), can then be written as

F(s)=∫∞0 f(t) e^−st dt

Provided that the integral exists. Where the Laplace Operator, s= σ + jω; will be real or complex j = √ −1

Properties of Transformation Calculator

Some of the Laplace transformation properties are:

If f1 (t) ⟷ F1 (s) and [note: ⟷ implies Laplace Transform]

f2 (t) ⟷ F2 (s), then,

Linearity PropertyA f1(t) + B f2(t) ⟷ A F1(s) + B F2(s)
Then, Frequency Shifting Propertye^s0t f(t)) ⟷ F(s – s0)
Then, Integrationt∫0 f(λ) dλ ⟷ 1⁄s F(s)
Multiplication by TimeT f(t) ⟷ (−d F(s)⁄ds)
Then, Complex Shift Propertyf(t) e^−at ⟷ F(s + a)
Then, Time Reversal Propertyf (-t) ⟷ F(-s)
Time Scaling Propertyf (t⁄a) ⟷ a F(as)

Transformation Calculator Table

The following Laplace transform table helps to solve the differential equations for different functions:

Sl No.f(t)L(f(t)) = F(s)Sl No.f(t)L(f(t)) = F(s)
11Similarly, 1/sSimilarly, 11e(at)Likewise, 1/(s − a)
Then, 2tn at t = 1,2,3,…Likewise, n!/s(n+1)Then, 12tp, at p>-1Likewise, Γ(p+1)/s(p+1)
Then, 3√(t)Likewise, √π/2s(3/2)Then, 13t(n-1/2) at n = 1,2,..Similarly, (1.3.5…(2n-1)√π)/(2n s(n+1/2)
Also, 4sin(at)Then, a/ (s2+a2)Likewise, 14cos(at)Similarly, s/(s²+a²)
Then, 5t sin(at)Then, 2as/ (s2+a2)2Then, 15t cos(at)Likewise, (s²-a²)/ (s²+a²)²
Then, 6sin(at+b)Likewise, (s sin(b)+ a cos(b) /(s2+a2)Also, 16cos(at+b)Likewise, (s cos(b)-a sin(b)/ (s²+a²)
Also, 7sinh(at)Likewise, a/ (s2-a2)Then, 17cosh(at)Similarly, s/(s²-a²)
Then, 8e^(at) sin(bt)Similarly, b/ ((s-a)² + b²)Then, 18e^(at) cos(bt)Similarly, (s-a)/ ( (s-a)² +b²)
Then, 9e^(ct) f(t)Similarly, F (s-c)Also, 19tnf(t) at n = 1,2,3Likewise, (-1)n Fn s
Also, 10f'(t)Likewise, sF (s)  – f(0)Then, 20f”(t)Likewise, s²F(s) − sf(0) − f'(0)

Transformation Calculator Differential Equation

A well-known mathematical technique for solving a differential equation is the Laplace transform. Transformations are used to address many mathematical issues. However, the objective is to turn the problem into a less difficult problem to tackle. However, the inverse transform, on the other hand, is useful for calculating the answer to a given issue.

Let us solve a first-order differential equation using the Laplace transformation for a better understanding.

Consider the equations y’-2y = e³x and y(0) = -5. Calculate the value of L. (y).

The linearity equation can be used to answer the first step of the equation:

L(y’ – 2y] = L(e3x)

L(y’) – L(2y) = 1/(s-3)

(because L(eax) = 1/(s-a))

L(y’) – 2s(y) = 1/(s-3)

sL(y) – y(0) – 2L(y) = 1/(s-3)

(Using Linearity property of the Laplace transform)

L(y)(s-2) + 5 = 1/(s-3) (Use value of y(0) ie -5 (given))

L(y)(s-2) = 1/(s-3) – 5

L(y) = (-5s+16)/(s-2)(s-3) …..(1)

here (-5s+16)/(s-2)(s-3) can be written as -6/s-2 + 1/(s-3) using partial fraction method

(1) implies L(y) = -6/(s-2) + 1/(s-3)

L(y) = -6e2x + e3x

Transformation Calculator With Points

However, the step function, often known as the Heaviside function, has the following definition:

If t is less than c, Uc(t) = 0-1; if t is more than or equal to c, Uc(t) = 0-1

However, the step function has two possible values: 0 and 1. It’s similar to an on/off switch. However, the Heaviside functions are denoted by the symbols uc(t), u(t-c), or H. (t-c)

Transformation Calculator Geometry

Geometric transformations are bijections that preserve geometric attributes, typically from the xy-plane to itself, although they can also be of higher dimensions. Meanwhile, in particular, there is only one real matrix representation for any linear geometric change. We can return a general transformation calculator for rotations, reflections, and shears, or we can compute the transformation matrix for a specific 2D or 3D transformation activity.

Transformation Calculator Bilateral Laplace

Bilateral Laplace transform is another name for the Laplace transform. This is also known as a two-sided Laplace transform, which is accomplished by expanding the integration limits to include the full real axis. As a result, the typical unilateral Laplace transform is transformed into a specific case of the Bilateral Laplace transform, with the function definition increased by the Heaviside step function.

However, the bilateral Laplace transform is defined as:

F(s) = ∫

However, the other way to represent the bilateral Laplace transform is B{F}, instead of F.

Transformation Calculator Graph

1. Identify The Parent Function

Everything you’re supposed to graph independently is predicated on a more fundamental graph (parent function) that you MUST know.

Look at the image above and ask your teacher which ones you’re in charge of.

2. Reflect Over X-Axis or Y-Axis

Reflect over the x-axis, or vertically, if there is a negative outside parentheses (all the y-values become negative)

f(x) = -x² is an example.

If a negative exists inside parenthesis, horizontally reflect over the y-axis (all x-values become negative) and factor out the negative.

ex: f(x) = 1 / (-x+3) becomes f(x) = 1 / (-(x-3))

3. Shift (Translate) Vertically or Horizontally

If a number is being added outside of parentheses, shift it up by that amount, or if a number is being removed, shift it down by that amount (ex: f(x) = x2–2 moves down 2).

Also, if a number is being added or subtracted inside parentheses, shift it left by that amount; if a number is being subtracted, move it right (opposite of what you’d anticipate).

f(x) = f(x) = f(x) = f(x) (x-2)

2 is moved 2 units to the right.

4. Vertical and Horizontal Stretches/Compressions

If there is a whole number coefficient outside of parentheses, multiply the y-values of all points by that coefficient to see the graph expand vertically, for example, f(x) = 4x² is stretched vertically by a factor of four.

If there is a fractional coefficient outside of parentheses, multiply the y-values of all points by that coefficient to see the graph compressed vertically, for example, f(x) = 1/2 x² compressed vertically by a factor of two.

Similarly, if a whole number coefficient is enclosed in parenthesis, multiply the x-values of all points by the inverse of (!) that coefficient to see the graph spread horizontally (contrary to expectations).

f(x) = f(x) = f(x) = f(x) (4x) Horizontally, 2 is compressed by a factor of four.

If a fractional coefficient is enclosed in parenthesis, multiply all point x-values by the inverse of (!) that coefficient to see the graph compress horizontally (the opposite of what you’d expect).

f(x) = (1/2 x — 4) is an example.

(1/2 (x — 8)) becomes 2.

By a ratio of two, 2 is expanded horizontally.

5. Plug in a couple of your coordinates into the parent function to double check your work

That means you can always double-check your work by entering an x-value (I recommend x=0) and seeing if the y-value matches your graph’s y-value.

Transformation Calculator Inverse Laplace

However, the inverse Laplace transform gives us the transform F(s) and asks us to figure out what function we started with. f(t) = L-1F(s)

For example, the inverse Laplace transform for two Laplace transforms, say F(s) and G(s), is defined as: L-1aF(s)+bG(s)= an L-1F(s)+bL-1 G(s).

However, the constants a and b are used in this equation.

In this scenario, we can take the inverse transform for each transform, add their constant values in their appropriate places, and run the operation to get the result.

Transformation Calculator Convolution Integrals

However, the convolution integral of f(t) and g(t) is given as: If the functions f(t) and g(t) are piecewise continuous functions on the interval [0,], then

(f * g) (t) =0t f(t-T) g(T)dT f(t-T) g(T)dT

As a result, the convolution integral follows the property (f*g)(t) = (g*) (t)

0t f(t-T) g(T)dT = 0t f(T) g(t-T)dt = 0t f(T) g(t-T)dt

As a result, the preceding fact will assist us in calculating the inverse transform of the sum of transforms.

(i.e.) L(f*g) = F(s) G(s)

L-1 {F(s)G(s)} = (f*g)(t).

Transformation Calculator Probability

However, the Laplace transform is defined as the expected value in both pure and applied probability theory. If X is a random variable with a probability density function, such as f, then the Laplace transform of f is: Lf(S) = E[e-sX], which is also known as the Laplace transform of random variable X.

Applications of Transformation Calculator

  • However, we use it to simplify complicated differential equations by using polynomials.
  • We use it to convert derivatives into numerous domain variables, then utilise the Inverse Laplace transform to convert the polynomials back to the differential equation.
  • Also, we use it in telecommunications to deliver signals to both sides of a media. When signals are conveyed over the phone, for example, first we transform them into a time-varying wave before we superimpose it on the medium.
  • We also used it for electrical circuit analysis, digital signal processing, system modelling, and other engineering jobs.

Real life applications

Lerch’s Cancellation Law is the source of Laplace’s Transform. However, the function in the time domain is changed into a Laplace function in the frequency domain using the Laplace Transform method. This Laplace function can be solved using an algebraic equation. However, the solution can be converted back to the time domain using an Inverse Laplace Transform.

This transform is most typically employed in control systems, as briefly explained above. However, the transforms are employed in the study and analysis of systems such as ventilation, heating, and air conditioning. These mechanisms are found in almost every modern structure.

Laplace transformations are also important for process control. They aid in the analysis of variables that, when altered, create the intended results. Experiments utilising heat are an example.

Laplace transforms are utilised in a wide range of technical applications, in addition to these two examples, and are quite beneficial. Both electronic and mechanical engineering benefit from them.

A differential equation can be used to model an electrical, mechanical, thermal, hydraulic, or other dynamic control system.

Transformation Calculator Matrix

Matrixes can be used to express the above transformations (rotation, reflection, scaling, and shearing). We multiply the transformation matrix by a column vector that represents the point coordinate to find the image of a point.

The following are the transformation matrices:

Type of transformationTransformation matrix
Then, Clockwise rotation by an angle θ about the origin| Cosθ  Sinθ |

| -Sinθ Cosθ |


Then, Counter-clockwise rotation by an angle θ about the origin| Cosθ   -Sinθ |

| Sinθ    Cosθ |

Also, Reflection against the x-axis| 1  0 |

| 0 -1 |

Similarly, Reflection against the y-axis| -1 0 |

|  0 1 |

Then, Scaling (contraction or dilation) in both x and y directions by a factor k| k 0 |

| 0 k |

Also, Horizontal shear (parallel to the x-axis) by a factor m| 1 m |

| 0 1 |

Then, Vertical shear (parallel to the y-axis) by a factor m| 1 0 |

| m 1 |

Transformation Calculator Linear Algebra

In mathematics, a transformation is the transformation of one function into another function that may or may not be in the same domain. We can use the transform approach to solve problems that cannot be solved directly. We named the Laplace transform after Pierre Simon Laplace, a French mathematician and astronomer.

He employed a similar transformation in his contributions to probability theory. It gained a lot of popularity after WWII. Oliver Heaviside popularised this transformation as an English electrical engineer. Niels Abel, Mathias Lerch, and Thomas Bromwich used it in the nineteenth century.

We may trace the history of the Laplace Transform all the way back to 1744. Another prominent mathematician, Leonhard Euler, was working on various sorts of integrals at the same time. He didn’t go very far with it, though, and he abandoned it. Joseph Lagrange, who refined Euler’s work and did additional study, was one of Euler’s admirers.

It became interested in LaGrange’s work 38 years later, in 1782, when he started up where Euler left off. Also, it had a stroke of genius in 1785 that forever transformed the way we solve differential equations. However, it worked on it until 1809, when he began utilising infinity as an integral condition, revealing the true power of the Laplace transform.

Disadvantages of the Transformation Calculator

We can only use the Laplace transform to solve complicated differential equations, and it, like other brilliant approaches, has a drawback that is not obvious. As a result, we can only solve differential equations with proper constants using this method. If you don’t have any exact constants in your equation, you’ll need to come up with another method.

Frequently Asked Questions on Transformation Calculator

What is the use of Laplace Transform?

We solve differential equations using the Laplace transform. Everyone accepted it widely in a variety of professions. However, the Laplace transform reduces a linear differential equation (LDE) to an algebraic equation, which may then be solved using basic algebraic identities.

How do you calculate Laplace transform?

However, the steps to calculate the Laplace transform are as follows:

Step 1: Divide the given function, f(t), by e-st, where s is a complex number and s = x + iy.

Step 2: Integrate this product with regard to time (t) using the 0 and 0 limitations.

We obtained the Laplace transformation of f(t) as a result of this procedure, that we denote with F. (s).

What is the Laplace method?

We named the Laplace transform (or Laplace technique) after Pierre Simon De Laplace, a prominent French mathematician (1749-1827). We use this method to determine the approximate value of a function’s integration. However, the Laplace transform transforms one signal into another using a set of predetermined rules or equations.

What are the properties of Laplace Transform?

However, the important properties of Laplace transform include:

Linearity Property: A f_1(t) + B f_2(t) ⟷ A F_1(s) + B F_2(s)

Frequency Shifting Property: es0t f(t)) ⟷ F(s – s0)

nth Derivative Property: (d^n f(t)/ dt^n) ⟷ s^n F(s) − n∑i = 1 s^{n − i} f^{i − 1} (0^−)

Integration: t∫_0 f(λ) dλ ⟷ 1⁄s F(s)

Multiplication by Time: T f(t) ⟷ (−d F(s)⁄ds)

Complex Shift Property: f(t) e^{−at} ⟷ F(s + a)

Time Reversal Property: f (-t) ⟷ F(-s)

Time Scaling Property: f (t⁄a) ⟷ a F(as)

What is the Laplace transform of sin t?

The Laplace transform of f(t) = sin t is L{sin t} = 1/(s^2 + 1). As we know, the Laplace transform of sin at = a/(s^2 + a^2).

How do you find the transformation of a function?

However, the function transformation takes whatever the underlying function f (x) is and “transforms” it, which is just a fancy way of stating that we changed the formula and moved the graph. This is a three-unit increase above the standard quadratic equation.f (x) = x². That is, x² + 3 is f (x) + 3.

How do you find the transformation of a graph?

Find out what the parent function is. Ernest Wolfe is a well-known author.

Over the X-Axis or the Y-Axis, reflect.

Vertically or horizontally shift (translate).

Stretches and compressions in both the vertical and horizontal planes.

To double-check your work, enter a handful of your coordinates into the parent function.

What is the transformation of a graph?

However, the graph transformation method entails altering an existing graph, or graphed equation, in order to generate variations of the original graph. We can extend, rotate, invert graphs or any combination of these transformations. We can also move them along the xy plane.

How do you describe transformation?

A transformation is a method of altering a shape’s size or position. We move every point in the shape in the same direction and at the same distance.

What is Transformation in quadratic equations?

You may sometimes observe how a quadratic function has been changed from the simple formula y=x² by looking at it. Then we can graph the equation by changing the “parent graph” in the appropriate way. For a positive value c, for example, the graph y=x²+c is the same when y=x² changes c units higher.

What is a transformation in a math example?

When we change a shape, size, or location in math, we call it a transformation. We can slide, flip, or turn Jigsaw puzzle pieces to change their positions. Each of these movements alters the piece’s form.