Describing curves using parameter-based functions
Parametric equations describe the coordinates of a point (x,y) in terms of a third variable called a parameter, usually denoted as t.
The parameter t acts as a control variable, determining both x and y coordinates as it varies.
Moving between parametric and rectangular forms
A circle with radius r can be parameterized as:
Square and add the equations:
Finding slopes and rates of change
The derivative dy/dx can be found using the chain rule:
This represents the slope of the tangent line at any point on the curve.
\[x = t^2\] \[y = t^3\]
\[\frac{dx}{dt} = 2t\] \[\frac{dy}{dt} = 3t^2\]
\[\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2}\]
Using parametric equations to model motion
\[x = v_0\cos(\theta)t\]
Where v₀ is initial velocity and θ is launch angle
\[y = v_0\sin(\theta)t - \frac{1}{2}gt^2\]
Where g is acceleration due to gravity (≈ 9.8 m/s²)
Test your understanding of parametric equations.