Parametric Equations

On your list of ‘to do’ things in this topic are parametric equations. Sweet little things, they involve a three way relationship with x, y and t. Something like x = 2t and y = t - 4 for 0 ≤ t ≤ 6 So you pop the values for t into the equations for x and y to give you some coordinates to plot. Dead easy. You can even do it with trigonometrical functions like x = cosθ y = sinθ 0 ≤ θ ≤ 360 What fun !

And then you end up with graphs like these:

Or this X= 2cos5θ y=2sinθ 0 ≤ t ≤ 360 which is much prettier

Here ends the first lesson Now you get to draw some yourself, but there aren’t very many pretty ones Sorry! The work can be found On a worksheet what I wrote for you especially. Porkess Pure 3 page 94 Exercise 4a 1(i, ii) -8 Pink Pure pg 327 q 26, 27, 28 Here ends the first lesson

The Joys of Eliminating Parameters Now for part two The Joys of Eliminating Parameters So what we do here is take the 2 equations in x and y, eliminate the t and end up with something just in x and y. Read this again . . . Lets look at x = 2t and y = t – 4 => t = y + 4 => x = 2(y + 4) => After this they only get harder . . . .

Find the cartesian equation of the curve whose parametric equations are: x = t² y = 2t 2. x = cosθ y = sinθ 3. x = 2t y = 2/t 1. y = 2t => t = ½y x = (½y)² x = ¼y² 2. Using cos²θ + sin²θ = 1 gives x² + y² = 1 y = 2/t => t = 2/y x = 2(2/y) x =

This ends part 2 go do some work . . . 1. Pink Pure page 327 questions 1 - 11 Porkess Pure 3 pg 94 question 1 (iii) Stanley Thornes pg 480 question 1

This next bit is all about differentiating parametric equations and then finding tangents, normals and stationary points yippee So to find y’ we use Using basic fraction rules gives So lets do some examples . . . .

Find the stationary point on the curve whose parametric equations are x = t³ y = (t+1)² so At a stationary point so t = -1 When t=-1 x=-1 y = 0 Therefore the stationary point is (-1, 0) Now you determine whether it’s a maximum or minimum . . . . . Click for answer minimum

Questions can be found in: Pink Pure pg 327 ex 13D questions 12 – 24 Stanley Thornes pg 480 questions 2 – 5 Porkess Pure 3 ex 4B pg 102