If x=3^t,y=81^t−32 are parametric equations of a curve, find a Cartesian equation of the curve. SHOW WORK.

81 = 3^4, so 81^t = (3^4)^t = 3^(4t) = (3^t)^4

y = x^4 - 32

Did you know?

Did you know that parametric equations can be converted into Cartesian equations to represent curves? Let's look at an example. Given the parametric equations: x = 3^t and y = 81^t - 32, we can find the corresponding Cartesian equation. To do this, we can eliminate the parameter t by solving for t in terms of x. From the first equation, we have t = log₃(x). Substituting this value of t into the second equation, we get y = (81^(log₃(x))) - 32. Using the property of logarithms, we can rewrite the equation as y = x² - 32, which is the Cartesian equation of the curve. So, there you have it! Parametric equations like x = 3^t and y = 81^t - 32 can be transformed into Cartesian equations, like y = x² - 32, to represent curves.