Geometry Parametric equations

Starter: challenge

Parametric equations: Cartesian equations KUS objectives BAT convert between parametric and Cartesian equations of a function Starter: previous page Geogebra: parametric eqns http://ggbtu.be/mni1QRU41

WB5 Parametric eqns – cartesian eqn example I Find the Cartesian equations of the following curves a) 𝑥=𝑡−1 𝑦= 2𝑡 2 +3𝑡 𝑡∈ℛ b) 𝑥= 𝑡 2 𝑦= 2𝑡 2 +3𝑡 𝑡≥0

WB6 find the Cartesian equations of these (eliminate the parameters) answers 𝑥=4𝑡 , 𝑦=3−𝑡 𝑥=𝑡−1 ,𝑦= 𝑡 2 +1 𝑥=2 𝑡 2 , 𝑦=4(𝑡−1) 𝑥=𝑡+2 , 𝑦= 1 𝑡 𝑥= 𝑡 2 −1 , 𝑦= 𝑡 2 +1 𝑥= 𝑡 2 −1 , 𝑦= 𝑡 4 +1 i) ii) iii) iv) v) vi)

WB 7 Draw the curve given by the Parametric Equations: x= 1 𝑡+1 𝑦= 𝑡 2 for −3≤𝑡≤3 -3 -2 -1 1 2 3 x = 1/(t+1) -1/2 -1  1 ½ 1/3 ¼ y = t2 9 4 1 1 4 9 Hmmmm… is this enough to sketch this graph? equation of the curve? 𝑥= 1 𝑡+1 𝑦= 𝑡 2 𝑦= 1 𝑥 −1 2 𝑥(𝑡+1)=1 𝑡+1= 1 𝑥 𝑦= 1 𝑥 − 𝑥 𝑥 2 𝑡= 1 𝑥 −1 𝑦= 1−𝑥 𝑥 2 𝑦= (1−𝑥) 𝑥 2 2

WB8 Trigonometric Parametric eqns – cartesian eqn Find the Cartesian equation of the following curve 𝑥= 𝑠𝑖𝑛 𝑡 𝑦= cos 𝑡 −𝜋≤𝑡≤ 𝜋

A curve has Parametric equations: 𝑥=𝑠𝑖𝑛𝑡+2 𝑦=𝑐𝑜𝑠𝑡−3 WB 9 A curve has Parametric equations: 𝑥=𝑠𝑖𝑛𝑡+2 𝑦=𝑐𝑜𝑠𝑡−3 a) Find the Cartesian equation of the curve b) Sketch the curve A Cartesian equation is just an equation of a line where the variables used are x and y only 𝑥=𝑠𝑖𝑛𝑡+2 𝑦=𝑐𝑜𝑠𝑡−3 𝑥−2=𝑠𝑖𝑛𝑡 𝑦+3=𝑐𝑜𝑠𝑡 How can we link sin t and cos t in an equation? 𝑠𝑖 𝑛 2 𝑡+𝑐𝑜 𝑠 2 𝑡≡1 (𝑥−2 ) 2 +(𝑦+3 ) 2 = 1 The equation is that of a circle  Think about where the centre will be, and its radius 5 (𝑥−2 ) 2 +(𝑦+3 ) 2 =1 Centre = (2, -3) Radius = 1 -5 5 -5

Another way of writing this (by squaring the whole of each side) WB 10 A curve has Parametric equations: 𝑥=𝑠𝑖𝑛𝑡 𝑦= sin 2𝑡 a) Find the Cartesian equation of the curve b) Sketch the curve 𝑥=𝑠𝑖𝑛𝑡 𝑦=𝑠𝑖𝑛2𝑡 double angle formula 𝑥 2 =𝑠𝑖 𝑛 2 𝑡 𝑦=2𝑠𝑖𝑛𝑡𝑐𝑜𝑠𝑡 Replace sint with x 𝑦=2𝑥𝑐𝑜𝑠𝑡 𝑠𝑖 𝑛 2 𝑡+𝑐𝑜 𝑠 2 𝑡≡1 b) Geogebra: parametric eqns 𝑐𝑜 𝑠 2 𝑡=1−𝑠𝑖 𝑛 2 𝑡 http://ggbtu.be/mni1QRU41 Replace sin2t with x2 𝑐𝑜 𝑠 2 𝑡=1− 𝑥 2 𝑐𝑜𝑠𝑡= 1− 𝑥 2 𝑦=2𝑥𝑐𝑜𝑠𝑡 We can now replace cos t 𝑦=2𝑥 1− 𝑥 2 Another way of writing this (by squaring the whole of each side) 𝑦 2 =4 𝑥 2 (1− 𝑥 2 )

WB11 find the Cartesian equations of these (eliminate the parameters) answers 𝑥=3 sin 𝑡 , 𝑦=2 cos 𝑡 𝑥= sec 𝑡 ,𝑦=5 tan 𝑡 𝑥=1+ cos 𝑡 , 𝑦=1−2 sin 𝑡 𝑖𝑣) 𝑥= cos 𝑡 + sin 𝑡 , 𝑦=2 cos 𝑡 + sin 𝑡 𝑣) 𝑥= cos 𝑡+ 𝜋 4 , 𝑦= 2 sin 𝑡 𝑣𝑖) 𝑥=2 cos 𝑡 −1 , 𝑦=3+2 sin 𝑡 i) ii) iii) iv) v) vi)

Parametric equations –Summary Parametric equations are written as: A Cartesian equation would be There are three main types of question in the exam Sketch a graph from parametric equations Eliminate (t) to find the Cartesian equation Differentiating to find gradients, tangents and normals Make a table of values – for t, x and y and plot points (x, y) ‘Zoom in’ on any interesting points – work out more values e.g. to check asymptotes are correct Write one thing you have learned Write one thing you need to improve