Parametric Equations – Easy to Calculate

When you hear about Parametric equations, what exactly comes to your mind? A mathematical calculation that is difficult? Well, we have good news. The parametric equation is not technical to compute. It is actually just the graphing of a continuous equation or equations based on a parameter t.

So why do we need to compute ordinary equations in the eye of a new variable known as a parameter? The simple answer is that the parameter t makes it much easier to compute certain problems whose solutions would be impossible to find without the parameter.

For example, parametric equations can be used to describe the curve of a Cartesian plane that, under normal circumstances, can’t be defined. Parametric equations are also used in solving issues relating to three-dimensional spaces and bigger dimensions. As such, it is important to understand more about the parameter t.

In general, it is important to know that the parameter t and its influence in making normal continuous equation parametric equations is not a problem you will need to deal with often in your mathematics journey. The calculations are extremely easy and straightforward, and the examples in this article will highlight that. Your focus should be on how understanding what it is as a theory and where it can be applied. All of these will be considered in this piece.

What Are Parametric Equations?

A parametric equation is a special equation with an independent variable called the parameter, denoted as t defines some dependent variables such that they become continuous functions.

The dependent variables are only defined by the parameter t and will move with respect to it.

For example, Consider the two variables

X = t

Y = 2t² + 3

X and Y, in this case, are the dependent variables, while t, which will eventually determine how the two variables will move, is the parameter.

The two dependent variables and the parameter are known as a parametric equation.

The values for t are often made available to make it much easier to fund the continuous function the two or more dependant variables represent.

In certain cases, though, more than one parameter may be available. You will only need to follow the same calculations as when dealing with just a parameter. No unusual calculation is needed.

Points to Consider When Computing the Parametric Equation

Some Points to Note when calculating a Parametric equation

• The parameter which is often represented with the symbol t often coverall a range of several numbers that also determines the values of the dependent variables.

• The parameter t, will generally determine the direction of the parametric equations represented by X and Y. if t moves in ascending order, then X and Y must move in the same order anytime t is substituted into them.

• Generally, you will need to handle two parametric equations represented with X and Y. However, it is essential to know that the major parametric equation is the one represented by Y. As such, for problems that involve eliminating the parameter t, the X variables will be substituted into the Y variable, and the graph will be plotted with respect to the Parametric equation represented by Y. This can be seen in example 3.

How to Calculate the Parametric Equation?

Example 1

Let us consider the parameter t on the basis of the two dependent variables, X and Y. The equations and parameters are given below

X = t + 3

y = t³

while we have the parameter in the form = – 2  t  2

Now find the variables related to the parameter t and plot all three of them.

Solution

Now, based on what we know, we will have to calculate the X and Y values based on the t.

Clearly, we can see that the values of t are actually from the range of – 2 to 2

As such;

t = – 2, – 1, 0, 1, 2

since X = t + 3, we will have the following

for t = – 2, X = – 2 + 3 = 1

for t = – 1,  X = – 1 + 3 = 2

for t = 0,  X = 0 + 3 = 3

for t = 1, X = 1 + 3 = 4

for t = 2, X = 2 + 3 = 5

Also, since y = t³

for t = – 2,  y = – 2³ = – 8

for t = – 1,  y = – 1³ = – 1

for t = 0, y = 0³ = 0

for t = 1, y = 1³ = 1

for t = 2, y = – 2³ = 8

so we have the following

t = – 2, – 1, 0, 1, 2

X = 1, 2, 3, 4, 5

Y = – 8, – 1, 0, 1, 8

So we have answered the first part of the question.

Now we will need to plot the graph of the two variables and the parameters.

t = – 2, – 1, 0, 1, 2

X = 1, 2, 3, 4, 5

Y = – 8, – 1, 0, 1, 8

The graph for the X and Y variables has been plotted. The line represents the parameter t. also, it is important to note that the parameter t is moving in an ascending order, which also affects the variables’ movement.

It is essential to check the movement of the parameter and ensure that its movement also influences that of the variables dependent on it, or the variables cannot be considered as parametric equations.

So the calculation for this particular example has been completed.

Example 2

So let us consider the following parameter t, which is given as the range t  0. The variables for which it is a parameter are given as

Y = 3t + 1

Find the values for the equations

Solution

It is important to note that the values of t were not specified. However, we definitely know that it starts with zero and can range up to positive infinity. This means that we can select any positive number for t.

However, we need to be more careful. Notice that we have the x = . So it is best we select positive values that are actually perfect squares so that we will not have decimals when we take their square roots.

So we will go for the following numbers. t = 0, 1, 4, 9, 16

For t = 16, y = 3(16) + 1 = 49

So we have our parametric equations given as

t = 0, 1, 4, 9, 16

x = 0, 1, 2, 3, 4

y = 1, 4, 13, 28, 49

We have found The parametric equations, and it is clear that as t increases, the two variables also increase. You can decide to graph it by following the same process we followed in the first example.

Now the next example will consider ways we can use to eliminate the parameter t.

Example 3

Consider the parametric equations and eliminate the parameter t.

X = 2t + 4

Y = 4t²

Solution

Now recall that we stated that if we are to eliminate the parameter t, we will need to set the parametric equation for x = 2t + 4 in a way that t will be the dependent variable and substitute it into the other parametric equation Y = 4t²

X = 2t + 4

Implies

X – 4 = 2t

So we have successfully eliminated the t and have a new parametric equation, the eqn (2).

Eliminating the parameter t in a parametric equation will generally leave students with just one dependent variable, which is a rule for that variable to be y. It is important to note that just as you can eliminate the parameter t in a parametric equation, you can add it to two normal equations and form a parametric equation.

The process of doing so is what is known as the parametrization of the two equations. Most experts simply call it parametrization for short.

The next example will fully consider all that parametrization is all about

Example 4

Let x be an independent variable and Y a dependent variable for an equation given as

Y = x² + 5

Solution

It is important to note that there is no parameter t in this equation Y = x² + 5.

So the plan is to make x = t …. (1)

And that will make the equation in our problem

Y = t² + 5 …. (2)

Now we have been able to find the x and Y variables which are

x = t

Y = t² + 5

However, it is important to understand that the parameter t value range will need to be determined by the student. In certain instances, the instruction might highlight the need for student to determine the range of the parameter t and graph the variable in its entirety.

Conclusion

Typically, a Parametric equation is expected to have at least two dependent variables and a parameter. of course, there are special instances that come with eliminating and substituting a parameter, but the conditions remain the same. The values for t are often made available to make it much easier to find the continuous function of the two or more dependent variables represents.

We have comprehensively highlighted what parametric equations are and what students can hope to face anytime it comes up. Several examples have been considered to explain the different conditions that come with parametric equations. You can continue to practice until you perfect all the conditions.