Question

In Problems 43 and 44, the parametric equations of four curves are given. Graph each of them, indicating the orientation.

$$\begin{gathered} C_1:x=t,y=t^2,-4\le t\le 4 \\ {C}_2:x=\cos t,y=1-{\sin }^{2}t,0\le t\le \mathrm{\pi } \\ {C}_3:x=e^t,y=e^{2t},0\le t\le \ln 4 \\ {C}_4:x=\sqrt{t},y=t,0\le t\le 16 \end{gathered}$$

Short answer

The graph of each figure is shown in respective figures.

Step-by-step solution

Step 1. Given Information

The given parametric equations are

$$\begin{gathered} C_1:x=t,y=t^2,-4\le t\le 4 \\ {C}_2:x=\cos t,y=1-{\sin }^{2}t,0\le t\le \mathrm{\pi } \\ {C}_3:x=e^t,y=e^{2t},0\le t\le \ln 4 \\ {C}_4:x=\sqrt{t},y=t,0\le t\le 16 \end{gathered}$$

Step 2. Graph of C1

We have $x=t^2,y=t^2,-4\le t\le 4$

The coordinates of ordered pairs are $\left(-4,16\right),\left(-2,4\right),\left(0,0\right),\left(2,4\right),\left(4,16\right)$.

Plot these points on the plane and join them to obtain the graph

Step 3. Graph of C2

We have $x=\cos t,y=1-{\sin }^{2}t,0\le t\le \mathrm{\pi }$

The coordinates are $\left(1,1\right),\left(\frac{\sqrt{2}}{2},\frac{1}{2}\right),\left(0,0\right),\left(-\frac{\sqrt{2}}{2},\frac{1}{2}\right),\left(-1,1\right)$

Plot these points on the plane and join them to obtain the graph

Step 4. Graph of C3

We have $x=e^t,y=e^{2t},0\le t\le \ln 4$

The coordinates are $\left(1,1\right),\left(2,4\right),\left(3,9\right),\left(4,16\right)$

Plot these points on the plane and join them to obtain the graph

Step 5. Graph of C5

We have $x=\sqrt{t},y=t,0\le t\le 16$

The coordinates are $\left(0,0\right),\left(2,4\right),\left(2\sqrt{2},8\right),\left(2\sqrt{3},12\right),\left(4,16\right)$

Plot these points on the plane and join them to obtain the graph