Eliminate the parameter t to find a Cartesian equation in the form x=f(y)
x(t)=4t^2
y(t)=4+2t
Well, let's get rid of t and find a Cartesian equation. We can start by solving the second equation for t:
4 + 2t = y
2t = y - 4
t = (y - 4)/2
Now, let's substitute this expression for t into the first equation:
x = 4((y - 4)/2)^2
Simplifying that a bit, we get:
x = 4(y^2 - 8y + 16)/4
x = y^2 - 8y + 16
So, the Cartesian equation in the form x = f(y) is:
x = y^2 - 8y + 16
Voila! We've eliminated the parameter t to find the Cartesian equation. Now we can move on to other fun-filled math problems.
To eliminate the parameter t and find a Cartesian equation in the form x = f(y), we need to solve for t in terms of y from the equation y(t) = 4 + 2t.
Starting with the given equation:
y = 4 + 2t
We subtract 4 from both sides:
y - 4 = 2t
Divide both sides by 2:
(t = (y - 4) / 2
Now we substitute this expression for t into the equation x(t) = 4t^2:
x = 4((y - 4) / 2)^2
Simplifying the expression:
x = 4(y - 4)^2 / 4
Canceling out the 4 in the numerator and denominator:
x = (y - 4)^2
Therefore, the Cartesian equation, in the form x = f(y), is x = (y - 4)^2.
To eliminate the parameter, we need to find an equation that only involves x and y without any mention of t. To do this, we need to solve one of the given equations for t and substitute it into the other equation.
Let's solve the second equation, y(t) = 4 + 2t, for t:
y = 4 + 2t
2t = y - 4
t = (y - 4)/2
Now, substitute this expression for t into the first equation, x(t) = 4t^2:
x = 4((y - 4)/2)^2
x = 4(y - 4)^2/4
x = (y - 4)^2
Therefore, the Cartesian equation in the form x = f(y) is x = (y - 4)^2.
t = (y-4)/2 from second equation
so
x = 4 [ (y-4)/2 ]^2
x = (y-4)^2
x = y^2 -8 y + 16