Question

Q. Is γ(t) = (t2, t4) a parameterization of the parabola y=x2?

Answer 1

Answer on Question #70821 – Math – Geometry

Question

Is $y(t) = (t^2, t^4)$ a parameterization of the parabola $y = x^2$?

Solution

Let. $y = x^{2}$

If $x = t$, then

$$\left\{ \begin{array}{l} y = t ^ {2} \\ x = t \end{array} \right., -\infty < t < +\infty$$

is a parameterization of the parabola $y = x^2$.

Hence

$$\left\{ \begin{array}{l} y = t ^ {4} \geq 0, \\ x = t ^ {2} \geq 0, \end{array} \right.$$

is not a parameterization of the parabola $y = x^2$ because this parametrization does not describe the branch of parabola $y = x^2$, where $x < 0$.

Answer:

$(t^2, t^4)$ is not a parameterization of the parabola $y = x^2$;

$(t, t^2)$ is a parameterization of the parabola $y = x^2$.

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