Seneca
Sign up

Parametric Equations

Maths: Edexcel A Level Pure Maths, topic 3.3.1

What are Parametric Equations?

Parametric equations define the coordinates of a graph as functions of a third variable, tt. Each coordinate of the graph is given by the point (x=f(t),y=g(t))(x=f(t),y=g(t)).

Definition

Definition

  • We can define a curve using the equations:
  • x=f(t)x = f(t) and y=g(t)y = g(t)
  • These are called parametric equations.
  • Parametric equations define a point on the curve in terms of a parameter tt.
  • You must specify the domain of tt that the curve exists for when defining a curve parametrically.
Example

Example

  • A curve has parametric equations:
  • x=t−1,  y=2t+4,  −3≤t≤2x = t-1,\; y = 2t+4, \;-3\leq t \leq 2
  • By substituting in values of tt in the domain given, we can find the coordinates of points on the curve.
Finding points on the curve

Finding points on the curve

  • For example, when t=−1t=-1:
  • x=−1−1=−2x = -1-1 = -2
  • y=(2×−1)+4=2y=(2\times -1) +4=2.
Conversion

Conversion

  • If a curve is only in terms of the variables xx and yy, we call this a Cartesian equation.
  • We can convert parametric equations into a Cartesian equation by rearranging for tt and substituting it in.
Example

Example

  • A curve has parametric equations:
  • x=t−1,  y=2t+4,  −3≤t≤2x = t-1,\; y = 2t+4, \;-3\leq t \leq 2
  • Rearranging the first equation, we get:
  • x=t−1→t=x+1x = t-1 \rightarrow t = x+1
Substitute in $$t$$

Substitute in tt

  • Substituting this into the equation for yy:
  • y=2(x+1)+4=2x+6y = 2(x+1)+4 = 2x+6
  • That means the Cartesian equation for this curve is y=2x+6y = 2x+6
  • However, we can't finish there, as the curve was specified for the region −3≤t≤2-3\leq t\leq2.
Find the domain

Find the domain

  • We need to substitute in for tt into the inequality that describes its domain:
  • −3≤x+1≤2-3\leq x+1\leq2
  • Subtracting one from each side gives:
  • −4≤x≤1-4\leq x\leq 1
Cartesian equation

Cartesian equation

  • So the curve with parametric equations:
  • x=t−1,  y=2t+4,  −3≤t≤2x = t-1,\; y = 2t+4, \;-3\leq t \leq 2
  • Is the same as the Cartesian equation:
  • y=2x+6,−4≤x≤1y = 2x+6, -4\leq x\leq1

Test yourself on Parametric Equations

  • A curve has the parametric equations:
    x=3t+2,  y=t2−1,  −3≤t≤2x = 3t+2,\;y=t^2-1,\;-3\leq t \leq 2
    What are the coordinates of the point with t=3t = 3?

  • A curve has the parametric equations:
    x=4t,  y=t3,  −t>1x = \frac{4}{t},\;y=t^3,\;-t \gt 1
    What are the coordinates of the point with t=2t = 2?

Practise this topic for free