2) Eliminate the parameter for the following parametric equation. Show all your work. Simplify as far as you can.
x(t)=2√t
y(t)=4t^2+3
sorry.
4((x/2)^2)^2 + 3 = 4(x^2/4)^2 + 3 = 4(x^4/16) + 3 = x^4/4 + 3
You are correct! I apologize for my mistake in the previous response. The correct elimination of parameter is as follows:
From the equation for x(t), we have:
t = (x/2)^2
Substituting this into the equation for y(t), we get:
y = 4t^2 + 3 = 4(x/2)^4 + 3
Simplifying this expression, we have:
y = (x^4)/4 + 3
Therefore, the equation in terms of x and y is:
y = (x^4)/4 + 3.
To eliminate the parameter for the given parametric equations, we need to express the variable t in terms of either x or y.
From the equation x(t) = 2√t, we can solve for t as follows:
x = 2√t
Divide both sides by 2 to isolate the square root term:
x/2 = √t
Square both sides to eliminate the square root:
(x/2)^2 = (√t)^2
x^2/4 = t
Now, we have t = x^2/4.
Next, let's express t in terms of y using the equation y(t) = 4t^2 + 3:
y = 4t^2 + 3
Substitute t with x^2/4:
y = 4(x^2/4)^2 + 3
Simplify:
y = x^4/4 + 3
Therefore, the eliminated equation is y = x^4/4 + 3.
We can eliminate the parameter by isolating t in one of the equations and substituting into the other. From the equation for x(t), we have:
t = (x/2)^2
Substituting this into the equation for y(t), we get:
y = 4((x/2)^2)^2 + 3
Simplifying this expression, we have:
y = 2x^2 + 3
Therefore, the equation in terms of x and y is:
y = 2x^2 + 3.