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Explain how the parameter 't' affects the direction of movement in parametric equations.
As 't' increases, the (x, y) point moves along the defined path. The rate of change of x(t) and y(t) determines the speed and direction.
How do transformations affect the unit circle in parametric form?
Shifting the center involves adding constants to x(t) and y(t). Changing the radius involves multiplying cos(t) and sin(t) by the radius. Rotation involves adding a constant to 't'.
Describe the role of the direction vector in parametrizing a line segment.
The direction vector (x2 - x1, y2 - y1) determines the slope and orientation of the line segment. It's scaled by the parameter 'k' to move from (x1, y1) to (x2, y2).
Explain how to determine the center and radius of a circle from its parametric equations.
In the equations $x(t) = a + rcos(t)$ and $y(t) = b + rsin(t)$, (a, b) is the center and 'r' is the radius.
What are parametric equations?
Equations expressing x and y in terms of a third variable (parameter), often 't'.
What is a parametrically defined circle?
A circle described by equations using a parameter to show movement around it.
What is a parametrically defined line?
A line described by equations using a parameter to show movement along it.
What is the parameter 't' often interpreted as?
Time, indicating how the (x, y) point moves as 't' changes.
What is a direction vector in the context of parametric lines?
A vector that indicates the direction of the line segment, calculated from two points.
What does the parameter 'k' represent in a parametric line segment?
A value between 0 and 1 that determines the position along the line segment.
How do you find the endpoints of a parametric line segment?
Substitute the extreme values of the parameter (usually 0 and 1) into the parametric equations for x and y.
How to determine the equation of a circle given its parametric equations?
1. Identify the center (a,b) and radius r from the equations x(t) = a + rcos(t) and y(t) = b + rsin(t). 2. Write the equation in the form (x-a)^2 + (y-b)^2 = r^2.
How to eliminate the parameter 't' in parametric equations of a circle?
Use the trigonometric identity $cos^2(t) + sin^2(t) = 1$. Solve for cos(t) and sin(t) in terms of x and y, then substitute into the identity.
How do you find the direction vector of a line segment given two points?
Subtract the coordinates of the initial point from the coordinates of the terminal point: $(x_2 - x_1, y_2 - y_1)$.