Parametric Equations - Sketching Graphs (Edexcel A-Level Mathematics): Revision Notes

Parametric equations define both $x$ and $y$ as functions of a parameter $t$:

$x = f(t), \quad y = g(t)$

Here, $t$ is the parameter, and as it varies, the pair ($x(t), y(t)$) traces out a curve in the $xy$-plane.

a) Choose a Range for $t$

• Determine an appropriate range for the parameter $t$, based on the problem or context.
• Sometimes the problem specifies a range, but if not, start with a range that covers typical values (e.g., $t \in [0, 2\pi]$ for trigonometric functions).

b) Calculate Corresponding $x$ and $y$ Values

• For several values of $t$ within the chosen range, calculate the corresponding values of $x(t)$ and $y(t)$.
• Create a table of values if needed to help organize your points.

c) Plot the Points

• On a Cartesian coordinate system, plot the points ($x(t), y(t)$) for each value of $t$.
• Pay attention to the order of plotting as $t$ increases, since this will show the direction of the curve.

d) Draw the Curve

• Connect the plotted points smoothly, considering the nature of the functions involved (e.g., whether they are linear, quadratic, sinusoidal).
• Indicate the direction of increasing $t$ with an arrow on the curve.

e) Identify Key Features

• Identify and mark any important features such as intercepts, maxima, minima, or points where the curve changes direction.
• If applicable, find where the curve crosses the axes by setting $x(t) = 0$ or $y(t) = 0$ and solving for $t$.

Example : Let's sketch the graph of the following parametric equations:

$x = \cos t, \quad y = \sin t$

where $t$ ranges from 0 to 2π.

Step-by-Step Solution:

1. Range of $t$:

• $t$ ranges from $0$ to $2\pi$.

2. Calculate $x(t)$ and $y(t)$ Values:

• For $t = 0$: $x(0) = \cos 0 = 1$, $y(0) = \sin 0 = 0$.
• For $t = \frac{\pi}{2}$: $\ x\left( \frac{\pi}{2}\right) = \cos\left ( \frac{\pi}{2}\right) = 0), \ (y\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) = 1$.
• For $t = \pi$: $x(\pi$) = $\cos \pi = -1$, $y(\pi$) = $\sin \pi = 0$.
• For $t = \frac{3\pi}{2}$: $x$($\frac{3\pi}{2}$) $= \cos(\frac{3\pi}{2}$) = 0, $y\left(\frac{3\pi}{2}\right) = \sin\left(\frac{3\pi}{2}\right) = -1$.
• For $t = 2\pi$: $x(2\pi$) =$\cos 2\pi = 1$, $y(2\pi)$ = $\sin 2\pi = 0$.

3. Plot the Points:

• Plot the points $(1, 0)$, $(0, 1)$, $(-1, 0)$, $(0, -1)$, and back to $(1, 0)$ on the Cartesian plane.

4. Draw the Curve:

• Connect the points smoothly, forming a circle.
• Indicate the direction of increasing $t$ (counter clockwise in this case).

5. Identify Key Features:

• The curve is a circle with radius 1, centred at the origin $(0, 0)$. Graph:

• The circle is traced counter clockwise as $t$ increases from $0$ to $2\pi$.

Example : Consider the parametric equations:

$x = 2\cos t, \quad y = \sin t$

where $t$ ranges from 0 to 2π.

Step-by-Step Solution:

6. Range of $t$:

• $t$ ranges from $0$ to $2\pi$.

7. Calculate $x(t)$ and $y(t)$ Values:

• For $t = 0$: $x(0) = 2\cos 0 = 2$, $y(0) = \sin 0 = 0$.
• For $t = \frac{\pi}{2}$: $x( \frac{\pi}{2}$) = $2\cos( \frac{\pi}{2}$) $= 0$, $y( \frac{\pi}{2}$) =$\sin(\frac{\pi}{2}$) $= 1$.
• For $t = \pi$: $x(\pi) = 2\cos \pi = -2$, $y(\pi) = \sin \pi = 0$.
• For $t = \frac{3\pi}{2}$: $x\left(\frac{3\pi}{2}\right) = 2\cos\left(\frac{3\pi}{2}\right) = 0$, $y\left(\frac{3\pi}{2}\right) = \sin\left(\frac{3\pi}{2}\right) = -1$.
• For $t = 2\pi$: $x(2\pi) = 2\cos 2\pi = 2$, $y(2\pi) = \sin 2\pi = 0$.

8. Plot the Points:

• Plot the points $(2, 0)$, $(0, 1)$, $(-2, 0)$, $(0, -1)$, and back to $(2, 0)$.

9. Draw the Curve:

• Connect the points smoothly, forming an ellipse.
• Indicate the direction of increasing $t$ (again, counter clockwise).

10. Identify Key Features:

• The ellipse is centred at the origin, with a horizontal axis of length 4 (from $-2$ to $2$) and a vertical axis of length 2 (from $-1$ to $1$). Graph:

• The ellipse is wider along the $x$-axis than the $y$-axis.

• Parametric equations provide a flexible way to describe curves that might be difficult to express with a single Cartesian equation.
• Sketching parametric curves involves calculating specific points for given values of t, plotting them, and smoothly connecting them while indicating the direction of motion.
• Understanding the nature of the functions $f(t)$ and $g(t)$ is crucial for anticipating the shape of the curve and its behaviour as $t$ varies.