In parametric equations, x and y are expressed as functions of the same parameter t.
What do f(t) and g(t) represent in parametric equations?
Functions of t
As t changes in the equations \begin{cases} x =ty= t^{2} \end{cases}, the point moves along a parabola y=x2 called its path
Parametric equations allow for easy representation of complex shapes that are difficult to express as explicit functions.
Match the curve with its parametric equations:
Circle ↔️ x=rcos(t),y=rsin(t)
Ellipse ↔️ x=acos(t),y=bsin(t)
Line ↔️ x=x0+at,y=y0+bt
Order the steps to describe the movement of a point along a curve defined by parametric equations as t changes:
1️⃣ Change the value of t
2️⃣ Calculate the new coordinates (x,y)
3️⃣ Plot the point in the plane
4️⃣ Repeat the process
To find dxdy from parametric equations, we use the formula dxdy=dx/dtdy/dt, which is called the chain rule
The parametric derivative dxdy gives the slope of the tangent line to the curve at a specific point.
The equation of the tangent line to a parametric curve at (x_{0}, y_{0})</latex> is y−y0=dxdy(t0)(x−x0), where dxdy(t0) is the slope at the parameter value t_{0}
The velocity vector in parametric equations is given by v=(dtdx,dtdy).
Steps to find the tangent line to the curve defined by x=t2 and y=2t at t=1:
1️⃣ Find the derivatives dtdx and dtdy
2️⃣ Calculate dxdy
3️⃣ Find the coordinates (x0,y0) at t=1
4️⃣ Write the tangent line equation
At t=1, the slope dxdy(1) for the curve defined by x=t2 and y=2t is 1
Parametric equations define the coordinates of a point in terms of a parameter t