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What is a parameter in parametric equations?
A third variable (often 't') used to define both x and y coordinates: x = f(t), y = g(t).
What are parametric functions?
A way to describe curves and surfaces in a 2D space using a set of equations with a third variable (parameter).
What does the domain of a parametric function define?
The range of values for the parameter 't', which determines the start and end points of the curve.
Explain how changing the parameter 't' affects a parametric curve.
Changing 't' traces the curve. Restricted domain of 't' means the graph has start and end points.
Why are parametric functions useful for representing complex shapes?
They can represent circles, ellipses, and other complex shapes with a consistent set of equations, offering flexibility.
Why are parametric functions useful for animations?
By changing the parameter 't', we can make the curve move dynamically, creating animations and interactive graphics.
How does the domain of the parameter 't' affect the graph of a parametric function?
The domain of 't' determines the start and end points of the curve. A restricted domain results in a segment of the curve.
How to eliminate the parameter 't' to find the Cartesian equation?
Solve one parametric equation for 't', then substitute into the other equation.
How to find the coordinates of a point on a parametric curve at a specific value of 't'?
Substitute the value of 't' into both parametric equations: x = f(t), y = g(t).
How to determine the start and end points of a parametric curve?
Use the domain restrictions on 't'. Plug the minimum and maximum values of 't' into the equations to find the (x, y) coordinates.