Sketching parametric curves

In a nutshell

The majority of parametric curves will not result in curves you can easily recognise and sketch. The best way to plot parametric curves is to substitute values of the parameter into each of the equations.

Sketching a parametric curve

Tabulate the values of $t, x$ and $y$ using the parametric equations, then plot points and sketch the curve.

Example 1

Draw the curve given by the parametric equations:

$x = 2\sin(t) + 3, y = 5\cos(t), 0 \le t \le 2\pi$ 

Choose values for $t$ , spanning the domain given in the question. For each value of $t$ , substitute to find the corresponding $x$ and $y$ values, which will be the coordinates of the points on the curve.

$t$	$0$	$\dfrac{\pi}{4}$	$\dfrac{\pi}{2}$	$\dfrac{3\pi}{4}$	$\pi$	$\dfrac{5\pi}{4}$	$\dfrac{3\pi}{2}$	$\dfrac{7\pi}{4}$	$2\pi$
$x=2\sin (t) + 3$	$3$	$4.41$	$5$	$4.41$	$3$	$1.59$	$1$	$1.58$	$3$
$y = 5 \cos( t)$	$5$	$3.54$	$0$	$-3.54$	$-5$	$-3.54$	$0$	$3.54$	$5$

Plot the points and draw a curve through the points. This graph is an ellipse.

Maths; Parametric equations; KS5 Year 13; Sketching parametric curves

Example 2

Draw the curve given by the parametric equations:

$x=2t-3, y = t^2-1, -2 \le t \le 2$ 

Choose values for $t$ , spanning the domain given in the question. For each value of $t$ , substitute to find the corresponding $x$ and $y$ values, which will be the coordinates of the points on the curve.

$t$	$-2$	$-1.5$	$-1$	$-0.5$	$0$	$0.5$	$1$	$1.5$	$2$
$x=2t-3$	$-7$	$-6$	$-5$	$-4$	$-3$	$-2$	$-1$	$0$	$1$
$y=t^2-1$	$3$	$\dfrac54$	$0$	$-\dfrac34$	$-1$	$-\dfrac34$	$0$	$\dfrac54$	$3$

Plot the points and draw a curve through the points.

Maths; Parametric equations; KS5 Year 13; Sketching parametric curves