Q.14 - Calculus

Question

Show that the parametrization $x=2t+1,y=4t^2-4$ for $t\in [-1,\infty )$ has the same graph as the one we plotted point by point in the reading.

Step-by-Step Solution

Answer

The required parameterization of the curves $x=2t+1,y=4t^2-4$ traces exactly the same curve as that of the equations $x=\sin t+1,y={\sin }^{2}t-4$.

Step: 1 Given information

Consider the parametric curves $x=2f+1,y=4s^2-4, t\in [-1,\infty ).$

Step; 2 Calculation

Consider the parametric curves $x=2t+1,y=4t^2-4,t\in [-1,\infty )$.

The objective is to parametrize the curves.

To parameterize the curves eliminate the parameter t from the parametric equations.

Take $x=2 t+1$

Add $-1$ on both sides of the equation.

$$\begin{gathered} x-1=2t+1-1 \\ x-1=2t+1-1 \\ x-1=2t \end{gathered}$$

Now divide the equation by 2 .

$$\begin{gathered} \frac{x-1}{2}=\frac{2t}{2} \\ \frac{x-1}{2}=\frac{2t}{2} \\ \frac{x-1}{2}=t \end{gathered}$$

Now substitute $t=\frac{x-1}{2}$ in $y=4t^2-4$ then the equation becomes,

$\begin{array}{rcl}y& =& 4{\left(\frac{x-1}{2}\right)}^2-4y\\ & =& 4\frac{(x-1{)}^2}{4}-4y\\ & =& 4\frac{(x-1{)}^2}{4}-4\end{array}$

On further simplification,

$$\begin{gathered} y=(x-1{)}^2-4 \\ y=x^2+1-2x-4\left[\text{ since }(x-1{)}^2=x^2-2x+1\right] \\ y=x^2-2x-3 \end{gathered}$$

Step: 3 Further simplification

Since the parametrization of the curves gives the same equation that of $x=\sin t+1,y={\sin }^{2}t-4.$

Now check the parametrization of the curves $x=\sin t+1,y={\sin }^{2}t-4$.

To parameterize the curves eliminate the parameter t from the parametric equations.

Take$x=\sin t+1$

Add $-1$ on both sides of the equation.

$$\begin{gathered} x-1=\sin t+1-1 \\ x-1=\sin t+1t-1 \\ x-1=\sin t \end{gathered}$$

Now substitute$\sin t=x-1$ in the equation $y={\sin }^{2}t-4$.

Then,

$$\begin{gathered} y=(x-1{)}^2-4 \\ y=x^2-2x+1-4 \\ y=x^2-2x-3 \end{gathered}$$

Thus, the parameterization of the curves $x=2t+1,y=4t^2-4$ traces exactly the same curve as that of the equations $x=\sin t+1,y={\sin }^{2}t-4$.

Hence the required explanation.