Video: Finding the Shape of a Curve Defined Parametrically

A particle moves along a curve defined by the parametric equations π‘₯ = 2^(𝑑) and 𝑦 = 8^(𝑑). 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 𝑑 and 𝑦 is equal to eight to the power 𝑑. What is the shape of the trajectory of the particle?

We’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 𝑑 and 𝑦 is equal to eight to the power of 𝑑. These will tell us the π‘₯-coordinate and the 𝑦-coordinate of our particle at the time 𝑑. 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 𝑑 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 won’t give us all of the information about the relationship between π‘₯ and 𝑦. To do that, we’re going to want to rewrite our equation in a Cartesian form. To do this, we’re going to want to write our parametric equations in the form 𝑦 is some function of π‘₯. Let’s take a look at our parametric equations. We have π‘₯ is some exponential function and 𝑦 is some exponential function. So let’s try doing this by using our laws of exponents. So let’s start with our parametric equations. We see that π‘₯ is equal to two to the power of 𝑑. One thing we could try doing is rewriting 𝑦 to be a function of two to the power of 𝑑. Then we could just substitute π‘₯ is equal to two to the power of 𝑑.

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

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

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