1. To create a graph of this curve, first set up a table of values. Since the independent variable in both [latex]x\left(t\right)[/latex] and [latex]y\left(t\right)[/latex] is t, let t appear in the first column. Then [latex]x\left(t\right)[/latex] and [latex]y\left(t\right)[/latex] will appear in the second and third columns of the table.
   

   t	[latex]x\left(t\right)[/latex]	[latex]y\left(t\right)[/latex]
   −3	−4	−2
   −2	−3	0
   −1	−2	2
   0	−1	4
   1	0	6
   2	1	8

   
   
   The second and third columns in this table provide a set of points to be plotted. The graph of these points appears in Figure 3. The arrows on the graph indicate the orientation of the graph, that is, the direction that a point moves on the graph as t varies from −3 to 2.
   

   

   Figure 3. Graph of the plane curve described by the parametric equations in part a.

2. To create a graph of this curve, again set up a table of values.
   

   t	[latex]x\left(t\right)[/latex]	[latex]y\left(t\right)[/latex]
   −2	1	−3
   −1	−2	−1
   0	−3	1
   1	−2	3
   2	1	5
   3	6	7

   
   
   The second and third columns in this table give a set of points to be plotted (Figure 4). The first point on the graph (corresponding to [latex]t=-2[/latex]) has coordinates [latex]\left(1,-3\right)[/latex], and the last point (corresponding to [latex]t=3[/latex]) has coordinates [latex]\left(6,7\right)[/latex]. As [latex]t[/latex] progresses from −2 to 3, the point on the curve travels along a parabola. The direction the point moves is again called the orientation and is indicated on the graph.
   

   

   Figure 4. Graph of the plane curve described by the parametric equations in part b.

3. In this case, use multiples of [latex]\frac{\pi}{6}[/latex] for t and create another table of values:
   

   t	[latex]x\left(t\right)[/latex]	[latex]y\left(t\right)[/latex]		t	[latex]x\left(t\right)[/latex]	[latex]y\left(t\right)[/latex]
   0	4	0		[latex]\frac{7\pi }{6}[/latex]	[latex]-2\sqrt{3}\approx -3.5[/latex]	2
   [latex]\frac{\pi }{6}[/latex]	[latex]2\sqrt{3}\approx 3.5[/latex]	[latex]2[/latex]		[latex]\frac{4\pi }{3}[/latex]	−2	[latex]-2\sqrt{3}\approx -3.5[/latex]
   [latex]\frac{\pi }{3}[/latex]	[latex]2[/latex]	[latex]2\sqrt{3}\approx 3.5[/latex]		[latex]\frac{3\pi }{2}[/latex]	0	−4
   [latex]\frac{\pi }{2}[/latex]	0	4		[latex]\frac{5\pi }{3}[/latex]	2	[latex]-2\sqrt{3}\approx -3.5[/latex]
   [latex]\frac{2\pi }{3}[/latex]	−2	[latex]2\sqrt{3}\approx 3.5[/latex]		[latex]\frac{11\pi }{6}[/latex]	[latex]2\sqrt{3}\approx 3.5[/latex]	2
   [latex]\frac{5\pi }{6}[/latex]	[latex]-2\sqrt{3}\approx -3.5[/latex]	2		[latex]2\pi[/latex]	4	0
   [latex]\pi[/latex]	−4	0

   
   
   The graph of this plane curve appears in the following graph.
   

   

   Figure 5. Graph of the plane curve described by the parametric equations in part c.

   This is the graph of a circle with radius 4 centered at the origin, with a counterclockwise orientation. The starting point and ending points of the curve both have coordinates [latex]\left(4,0\right)[/latex].

1. To eliminate the parameter, we can solve either of the equations for t. For example, solving the first equation for t gives
   
   [latex]\begin{array}{ccc}\hfill x& =\hfill & \sqrt{2t+4}\hfill \\ \hfill {x}^{2}& =\hfill & 2t+4\hfill \\ \hfill {x}^{2}-4& =\hfill & 2t\hfill \\ \hfill t& =\hfill & \frac{{x}^{2}-4}{2}.\hfill \end{array}[/latex]

   
   
   Note that when we square both sides it is important to observe that [latex]x\ge 0[/latex]. Substituting [latex]t=\frac{{x}^{2}-4}{2}[/latex] this into [latex]y\left(t\right)[/latex] yields
   

   [latex]\begin{array}{ccc}\hfill y\left(t\right)& =\hfill & 2t+1\hfill \\ \hfill y& =\hfill & 2\left(\frac{{x}^{2}-4}{2}\right)+1\hfill \\ \hfill y& =\hfill & {x}^{2}-4+1\hfill \\ \hfill y& =\hfill & {x}^{2}-3.\hfill \end{array}[/latex]

   
   
   This is the equation of a parabola opening upward. There is, however, a domain restriction because of the limits on the parameter t. When [latex]t=-2[/latex], [latex]x=\sqrt{2\left(-2\right)+4}=0[/latex], and when [latex]t=6[/latex], [latex]x=\sqrt{2\left(6\right)+4}=4[/latex]. The graph of this plane curve follows.
   

   

   Figure 7. Graph of the plane curve described by the parametric equations in part a.

2. Sometimes it is necessary to be a bit creative in eliminating the parameter. The parametric equations for this example are
   
   [latex]x\left(t\right)=4\cos{t}\text{and}y\left(t\right)=3\sin{t}[/latex].

   
   
   Solving either equation for t directly is not advisable because sine and cosine are not one-to-one functions. However, dividing the first equation by 4 and the second equation by 3 (and suppressing the t) gives us
   

   [latex]\cos{t}=\frac{x}{4}\text{and}\sin{t}=\frac{y}{3}[/latex].

   
   
   Now use the Pythagorean identity [latex]{\cos}^{2}t+{\sin}^{2}t=1[/latex] and replace the expressions for [latex]\sin{t}[/latex] and [latex]\cos{t}[/latex] with the equivalent expressions in terms of x and y. This gives
   

   [latex]\begin{array}{ccc}\hfill {\left(\frac{x}{4}\right)}^{2}+{\left(\frac{y}{3}\right)}^{2}& =\hfill & 1\hfill \\ \hfill \frac{{x}^{2}}{16}+\frac{{y}^{2}}{9}& =\hfill & 1.\hfill \end{array}[/latex]

   
   
   This is the equation of a horizontal ellipse centered at the origin, with semimajor axis 4 and semiminor axis 3 as shown in the following graph.
   

   

   Figure 8. Graph of the plane curve described by the parametric equations in part b.

   As t progresses from [latex]0[/latex] to [latex]2\pi[/latex], a point on the curve traverses the ellipse once, in a counterclockwise direction. Recall from the section opener that the orbit of Earth around the Sun is also elliptical. This is a perfect example of using parameterized curves to model a real-world phenomenon.

First, it is always possible to parameterize a curve by defining [latex]x\left(t\right)=t[/latex], then replacing x with t in the equation for [latex]y\left(t\right)[/latex]. This gives the parameterization

Since there is no restriction on the domain in the original graph, there is no restriction on the values of t.

We have complete freedom in the choice for the second parameterization. For example, we can choose [latex]x\left(t\right)=3t - 2[/latex]. The only thing we need to check is that there are no restrictions imposed on x; that is, the range of [latex]x\left(t\right)[/latex] is all real numbers. This is the case for [latex]x\left(t\right)=3t - 2[/latex]. Now since [latex]y=2{x}^{2}-3[/latex], we can substitute [latex]x\left(t\right)=3t - 2[/latex] for x. This gives

Therefore, a second parameterization of the curve can be written as