# Question Video: Vector-Valued Functions Mathematics • Higher Education

Sketch the graph of the vector-valued function π(π‘) = (π‘Β³) π + (π‘) π.

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### Video Transcript

Sketch the graph of the vector-valued function π of π‘ equals π‘ cubed π plus π‘ π.

Letβs begin by recalling what this vector-valued function actually tells us. It takes a real number π‘, and it outputs a position vector. Itβs horizontal component is π‘ cubed and its vertical component is π‘. So we could say that in the π₯π¦-plane, the π₯-value of any coordinate on our graph would be given by π‘ cubed and the π¦-value would be given by π‘. And there are two things we could do next. We could try forming a table and inputting values of π‘ and plotting the π₯- and π¦-coordinates. Alternatively, we can manipulate our equations for π₯ and π¦ to see if we can eliminate our parameter and get something we recognise. Letβs look at that latter method.

We are told that π¦ is equal to π‘. We can, therefore, replace π‘ with π¦ in our equation for π₯. And we find that π₯ is equal to π¦ cubed. We could alternatively say that π¦ is equal to the cube root of π₯. So how do we sketch this graph? Well, we know how to sketch the graph of π¦ equals π₯ cubed. But we also know that π¦ is equal to the cube root of π₯ is the inverse function of π¦ equals π₯ cubed. We then recall that to sketch the graph of an inverse function, we reflect the graph of the original function in the line π¦ equals π₯. And we obtain the graph of π¦ is equal to the cube root of π₯ or π₯ is equal to π¦ cubed, as shown.

Now, of course, we arenβt actually quite finished. Remember, we created a pair of parametric equations. And we know that when we plot a parametric graph, we must consider the direction in which the curve is sketched. So letβs take a couple of values. Letβs consider π‘ equals zero and π‘ equals one. When π‘ is equal to zero, π₯ is equal to zero cubed or zero and π¦ is equal to zero. Similarly, when π‘ is equal to one, π₯ is equal to one cubed, which is, of course, one and π¦ is also equal to one. So we start at the point zero, zero and we move up to the point one, one. This means weβre moving on this curve from left to right. And now, weβve finished. Weβve sketched the graph of the vector-valued function π of π‘ equals π‘ cubed π plus π‘ π.