x(t)=5sqrt(t) y(t)=4t+6 Find an equation for y as a function of x

Answer 1

If x = 5√t then x/5 = √t or t = x²/25

If t = x²/25 then wherever we see a "t" in the equation y = 4t + 6, we can plug in "x²/25"

So y = 4(x²/25) + 6. This is now an equation of y as a function of x and is written as:

y(x) = 4x²/25 + 6

To truly match the parametric equations, we would need to restrict the domain of x to be greater than or equal to zero since the domain of "t" is also greater than or equal to zero.

Answer 2

Hi Cady!

Solving this problem comes down to eliminating the parameter t. Basically we take one of the equations and solve it for t, and then plug the expression we get in for t in the other equation. Since we are going for y as a function of x, we need to solve the x(t) equation for t.

x(t)=5sqrt(t)

x=5sqrt(t) (Replace x(t) with x)

x/5=5sqrt(t)/5 (Divide both sides by 5)

x/5=sqrt(t) (Cancel the 5's off the right side)

(x/5)^2=sqrt(t)^2 (Square both sides)

Since squaring cancels a square root, we have t=(x/5)^2, which can be simplified to t=x^2/25.

Now plug x^2/25 in for t in the equation for y (make sure to change y(t) to y when doing so):

y=4(x^2/25)+6

There is really no more simplifying to do except dropping the parentheses, so y=4x^2/25+6. Now we have y as a function of x.