Question Video: Converting Parametric Equations into Rectangular Form Mathematics • 12th Grade

Convert the parametric equations 𝑥 = √𝑡 and 𝑦 = 5𝑡⁴ + 4𝑡 to rectangular form.

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Video Transcript

Convert the parametric equations 𝑥 is equal to the square root of 𝑡 and 𝑦 is equal to five 𝑡 to the fourth power plus four 𝑡 to rectangular form.

We’re given a pair of parametric equations. And we’re asked to convert these into the rectangular form. Recall, this means we need to convert this into an equation involving just 𝑥 and 𝑦. We need to eliminate the variable 𝑡. And there’s lots of different methods we can try for different types of parametric equations. Usually, the best way to start is to take a look at our parametric equations.

In this instance, we can see 𝑥 is equal to the root of 𝑡 and 𝑦 is equal to five 𝑡 to the fourth power plus four 𝑡. And we want to eliminate the variable 𝑡 from this expression. And we can see that 𝑥 is equal to the root of 𝑡. We could just substitute this into our expression for 𝑦. Recall, if 𝑦 is five 𝑡 to the fourth power plus four 𝑡, then we can replace 𝑡 in this expression with the square root of 𝑡 squared. So 𝑦 is equal to five times root 𝑡 all squared all raised to the fourth power plus four times root 𝑡 all squared.

But remember, root 𝑡 is equal to 𝑥. So we could replace each of these expressions of root 𝑡 with 𝑥. Doing this gives us five times 𝑥 squared all raised to the fourth power plus four 𝑥 squared. Then we can simplify this. By using our laws of exponents, 𝑥 squared all raised to the fourth power is equal to 𝑥 to the power of two times four. And then we can evaluate this to give us our final answer: 𝑦 is equal to five 𝑥 to the eighth power plus four 𝑥 squared.

Therefore, we were able to convert the parametric equations 𝑥 is equal to root 𝑡 and 𝑦 is equal to five 𝑡 to the fourth power plus four 𝑡 to the rectangular form. We got 𝑦 is equal to five 𝑥 to the eighth power plus four 𝑥 squared.