Question

Consider the provided statement to sketch the parametric curve and then eliminate the parameter to find Cartesian equation of the curve.

X = eᵗ, y=1-e³ᵗ, 0

Answer 1

Considering the provided statement to sketch the parametric curve, Cartesian equation of the curve. x = eᵗ, y = 1 - e³ᵗ, is y = 1 - x³

Considering the provided statement to sketch the parametric curve and then eliminate the parameter to find Cartesian equation of the curve. x = eᵗ, y = 1 - e³ᵗ, we proceed as follows

Since we have the parametric equations x = eᵗ, y = 1 - e³ᵗ, re-writing the second equation, we have that

y = 1 - e³ᵗ

y - 1 = - e³ᵗ

e³ᵗ = -(y - 1)

e³ᵗ = 1 - y

Now, since x = eᵗ, talking the cube of both sides, we have that

x = eᵗ

x³ = (eᵗ)³

x³ = e³ᵗ

Now, equating both expressions, we have that

e³ᵗ = 1 - y

x³ = 1 - y

Re-arranging, we have that

y = 1 - x³

So, the cartesian equation is y = 1 - x³