Find two different sets of parametric equations for the following equation: y= (1/(x^3)) I need the answers to this.

Question

I have the answers {x=2t, y=(1/8t^3) } and {x=t, y=(1/t^3)}

If I have the wrong answers could you explain how to get the right ones? I have to answer this in class tomorrow and I'm insanely paranoid that it's wrong and I'll make a fool of myself.

Answer 1

If you have an equation y = f(x) then you can define x=g(t), in which case substituting into the equation for y, you get y=f(g(t)). The only constraint on the choice on the function g(t) is that the range of g(t) includes the entire domain of f(x). That way the graph of the parametric equations is the entire graph, not a partial one, of the original equation.

In your case, the two answers you gave are correct, as are many others: y = 1/x³ = 1/(2t)³ = 1/(8t³) in the first pair of equations, and y = 1/x³ = 1/t³ in the second pair of equations.

Good job.

In the future, don't be afraid if you are not sure of an answer. Even if you were to have it wrong, teachers and tutors are here to help you learn from mistakes, not criticise your abilities/intelligence. In our minds, you're a smart person.

Answer 2

Ginelle,

It's been awhile since I've done parametric equations, and with that disclaimer out of the way, here are my thoughts: 

I agree with the {x=t, y=(1/t³)} . I'm thinking the second one could be {(1/x)=t, y=t³)}. 

With x = 2t, you could then include additional possibilities of x= 3t, x = 4t, and so on.