7) Eliminate the parameter for the following parametric equations. Show all your work. Simplify as far as you can.
x(t) = 2t
y(t) = -4t^2+2
Nope, the bot screws up again.
the first part is right:
t = x/2
then in the 2nd equation:
y = -4t^2 + 2
= -4(t^2/4) + 2
= -t^2 + 2
My apologies, you are correct. Here is the corrected solution:
From the first equation, we have:
t = x/2
Substituting this into the second equation:
y = -4t^2 + 2
y = -4(x/2)^2 + 2
y = -4x^2/4 + 2
y = -x^2 + 2
Thus, we have eliminated the parameter t and expressed y solely in terms of x.
Final answer: y = -x^2 + 2
To eliminate the parameter t, we need to express t in terms of x or y and substitute it into the other equation. Let's start with the x(t) equation.
Given: x(t) = 2t
To eliminate the parameter t, we can solve for t.
Divide both sides by 2:
x(t)/2 = t
Now, we can substitute this expression for t into the y(t) equation:
y(t) = -4t^2 + 2
Replacing t with x(t)/2, we have:
y(x(t)/2) = -4(x(t)/2)^2 + 2
Simplifying further:
y(x/2)^2 = -4(x^2/4) + 2
Simplifying the expression in the parentheses:
y(x/2)^2 = -x^2 + 2
Therefore, eliminating the parameter t, we obtain the equation:
y = -x^2 + 2
From the first equation, we have:
t = x/2
Substituting this into the second equation:
y = -4(x/2)^2 + 2 = -2x^2 + 2
Thus, we have eliminated the parameter t and expressed y solely in terms of x.
Final answer: y = -2x^2 + 2