Parametric Equations
Definition of a Parametric Equation
● Cartesian equation - y = f (x)
● Parametric equation - x = f (t), y = g(t)
● Both x and y are defined in terms of a third variable (usually t or θ )
● Parametric equations can be used for a complicated curve which doesn’t
have a simple cartesian equation
● When sketching a parametric curve, find the x and y coordinates of the
values of the third variable, plot these points and join them
Cartesian Equation of a Parametric Curve
● Need to eliminate the third variable (e.g t or θ )
● Three methods:
○ Make t the subject of the x and y equations and substitute
○ Add/subtract the x and y equations so that it is easy to make t the
subject, then substitute and simplify
○ If the parametric equations involve trigonometric functions, find the
identity which connects the two trigonometric functions and
substitute
● Parametric equations can describe circles using trigonometry:
○ Circle with radius r and centre (0, 0) has parametric equations
x = r cos θ
y = r sin θ
○ Circle with radius r and centre (a, b) has parametric equations
x = a + r cos θ
y = b + r sin θ
Parametric Differentiation
Finding the Gradient of a Parametric Curve
dy dy
● Using the chain rule, we know that dx = dt × dt
dx
● Therefore we can find the gradient function of the curve by differentiating
the two parametric equations and combining as per the formula above
● This only works if dx
dt =/ 0
Definition of a Parametric Equation
● Cartesian equation - y = f (x)
● Parametric equation - x = f (t), y = g(t)
● Both x and y are defined in terms of a third variable (usually t or θ )
● Parametric equations can be used for a complicated curve which doesn’t
have a simple cartesian equation
● When sketching a parametric curve, find the x and y coordinates of the
values of the third variable, plot these points and join them
Cartesian Equation of a Parametric Curve
● Need to eliminate the third variable (e.g t or θ )
● Three methods:
○ Make t the subject of the x and y equations and substitute
○ Add/subtract the x and y equations so that it is easy to make t the
subject, then substitute and simplify
○ If the parametric equations involve trigonometric functions, find the
identity which connects the two trigonometric functions and
substitute
● Parametric equations can describe circles using trigonometry:
○ Circle with radius r and centre (0, 0) has parametric equations
x = r cos θ
y = r sin θ
○ Circle with radius r and centre (a, b) has parametric equations
x = a + r cos θ
y = b + r sin θ
Parametric Differentiation
Finding the Gradient of a Parametric Curve
dy dy
● Using the chain rule, we know that dx = dt × dt
dx
● Therefore we can find the gradient function of the curve by differentiating
the two parametric equations and combining as per the formula above
● This only works if dx
dt =/ 0
