# Question Video: Finding the Shape of a Curve Defined Parametrically Mathematics • Higher Education

A particle moves along a curve defined by the parametric equations π₯ = 2^(π‘) and π¦ = 8^(π‘). What is the shape of the trajectory of the particle?

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### Video Transcript

A particle moves along a curve defined by the parametric equations π₯ is equal to two to the power of π‘ and π¦ is equal to eight to the power π‘. What is the shape of the trajectory of the particle?

Weβre told that a particle is moving along a curve defined by a pair of parametric equations, π₯ is equal to two to the power of π‘ and π¦ is equal to eight to the power of π‘. These will tell us the π₯-coordinate and the π¦-coordinate of our particle at the time π‘. We need to use these parametric equations to find the shape of the trajectory of our particle. We might be tempted to do this by substituting values of π‘ into our pair of parametric equations. We could then plot the coordinates of this onto our graph, and it would give us a rough idea of the trajectory of our particle.

However, this wonβt give us all of the information about the relationship between π₯ and π¦. To do that, weβre going to want to rewrite our equation in a Cartesian form. To do this, weβre going to want to write our parametric equations in the form π¦ is some function of π₯. Letβs take a look at our parametric equations. We have π₯ is some exponential function and π¦ is some exponential function. So letβs try doing this by using our laws of exponents. So letβs start with our parametric equations. We see that π₯ is equal to two to the power of π‘. One thing we could try doing is rewriting π¦ to be a function of two to the power of π‘. Then we could just substitute π₯ is equal to two to the power of π‘.

To do this, weβll start by noticing that eight is equal to two cubed. This means we can rewrite π¦ as two cubed all raised to the power of π‘. To write this as a function of two to the power of π‘, weβre going to need to use our laws of exponents. First, we know π to the power π all raised to the power of π is just equal to π to the power of π times π. So using this, we get π¦ is equal to two to the power of three π‘. But remember, weβre trying to write π¦ in terms of two to the power of π‘. And to do this, weβll use this same rule again. However, this time weβll switch the values of π and π around. This means we can rewrite π¦ as two to the power of π‘ all cubed.

Now, all we need to do is use our substitution π₯ is equal to two to the power of π‘. And if we do this, we see that we get π¦ is equal to π₯ cubed. And remember, we were specifically asked about the shape of our trajectory. Well, if it follows the curve π¦ is equal to π₯ cubed, we can say that the trajectory follows a cubic curve. Therefore, weβve shown if a particle moves along a curve defined by the parametric equations π₯ is equal to two to the power of π‘ and π¦ is equal to eight to the power of π‘, then the trajectory of our particle follows the shape of a cubic curve.