Eliminate the parameter to find a cartesian equation
x= 1-t^2
y=t-2
-2 _<t _<2
known: -4 <(or=) y <(or =) 0
I got y= sq(x-1)+2
Close.
Rearrange x = 1 - t^2:
t = ±√(1-x)
Substitute into: y = t - 2
y = -2 ± √(1-x)
--Alternatively--
Rearrange : y = t - 2
t = y + 2
Substitute into x = 1 - t^2
x = 1 - (y+2)^2
Thus:
x = -3 - 4y - y^2
To eliminate the parameter and find the Cartesian equation, you need to express one variable in terms of the other variable. In this case, you want to express y in terms of x.
From the given parametric equations:
x = 1 - t^2 (Equation 1)
y = t - 2 (Equation 2)
Let's solve Equation 2 for t:
t = y + 2 (Equation 3)
Substitute Equation 3 into Equation 1:
x = 1 - (y + 2)^2
Simplify the equation:
x = 1 - (y^2 + 4y + 4)
x = 1 - y^2 - 4y - 4
x = -y^2 - 4y - 3
Now, let's consider the given constraint -4 ≤ y ≤ 0. We can observe that the equation represents a downward-opening parabola. The value of y ranges from -4 to 0, meaning the parabola has a vertex at the maximum y-value (-4) and opens downward.
To complete the process, let's rewrite the equation with x isolated:
x = -(y^2 + 4y + 3)
Thus, the Cartesian equation that eliminates the parameter is:
x = -(y^2 + 4y + 3)