P2.8 Parametric Equations

Question 1

how is a curve defined using parametric equations?

Answer 1

x- & y-coordinates of each point on a curve are functions of a third variable
x = p(t)
y = q(t)
each value of the parameter t defines a point on the curve

Question 2

how do you convert between parametric & Cartesian equations?

Answer 2

use substitution to eliminate the parameter t
Cartesian equation in 2d involves only x & y

Question 3

what is the domain & range of Cartesian equation y=f(x) formed from parametric equations x=p(t) & y=q(t)?

Answer 3

domain of f(x) is range of x=p(t)
range of f(x) is the range of y=q(t)
always sketch the Cartesian graph!

Question 4

domain & range graphically

Answer 4

sketch graph

Question 5

how do you convert parametric equations to Cartesian equation using trig. identities?

Answer 5

straight substitution - sub t with the trig.
double angle identity - common step to link sint with cost sin^2t + cos^2t = 1
Pythagorean identity

Question 6

sketching parametric curves

Answer 6

make table of t, x & y
substitute the values of the t into each equation to get x & y
plot x & y & always label t at each coordinate

Question 7

points of intersection questions - crossing x- or y- axis or a given line

Answer 7

find the value of the independent variable (usually t) at the point of interest
x- axis: set y=0
y-axis: set x=0

Question 8

how do you differentiate parametric equations?

Answer 8

1. find Cartesian then differentiate normally
   but be aware of the restraints on t!
2. chain rule
   dy/dx = dy/dt x dt/dx

Question 9

tips for parametric modelling questions

Answer 9

think what you are trying to achieve & how
what are the units?
does the answer make sense numerically (e.g. too big, too small, not negative time or length)
what is the significance of where the function crosses the x or y-axes?
what is special about any turning points?
what part of mathematics is it asking about? relate to what you have studied

Question 10

parametric stuff always in RADIANS