Extending what you know…

Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the equation To be able to convert a parametric equation to a Cartesian equation

Parametric Equations You are used to equations in Cartesian form. Eg y=f(x) Parametric equations are where x and y are defined by some other variable. Usually the letter t, y=f(t) x=g(t) Sometimes this is more useful and allows you to define more complex graphs.

Using Your Calculator You can plot parametric equations on your calculator. Enter graphing mode Press (F3) for TYPE And press (F3) again for Parm This will then change the display to show Xt1= Yt1= And you can enter the equations for x and y in terms of t.

Eliminating the Parameter Sometimes it is useful to eliminate the parameter. To do this you must rearrange the equations to be able to eliminate t. When the parametric equations contain sin and cos you may need to use trig identities to get an equation in a more useful form.

Examples Find the equation in Cartesian form from the parametric equations x = t + 1 and y = t2

Examples Define x=3cost and y=5sint in Cartesian form.

Pair up cards into Parametric and matching Cartesian form Card Match Pair up cards into Parametric and matching Cartesian form

Bilborough College Maths - core 4 parametric equations (Adrian) 24/04/2017 Sketch the curve with equations Solution: Let’s see how to do it without eliminating the parameter. We can easily spot the min and max values of x and y: ( It doesn’t matter that we don’t know which angle q is measuring. ) For both and the min is -1 and the max is +1, so and It’s also easy to get the other coordinate at each of these 4 key values e.g.

Bilborough College Maths - core 4 parametric equations (Adrian) 24/04/2017 and x We could draw up a table of values finding x and y for values of q but this is usually very inefficient. Try to just pick out significant features. x x x

Bilborough College Maths - core 4 parametric equations (Adrian) 24/04/2017 and x We could draw up a table of values finding x and y for values of q but this is usually very inefficient. Try to just pick out significant features. x x x x Think what happens to and as q increases from 0 to . This tells us what happens to x and y.

Bilborough College Maths - core 4 parametric equations (Adrian) 24/04/2017 and x Symmetry now completes the diagram. x x x x Think what happens to and as q increases from 0 to . This tells us what happens to x and y.

Bilborough College Maths - core 4 parametric equations (Adrian) 24/04/2017 and x Symmetry now completes the diagram. x x x

Bilborough College Maths - core 4 parametric equations (Adrian) 24/04/2017 and x Symmetry now completes the diagram. x x x

Bilborough College Maths - core 4 parametric equations (Adrian) 24/04/2017 The following equations give curves you need to recognise: a circle, radius r, centre the origin. a parabola, passing through the origin, with the x-axis as an axis of symmetry. an ellipse with centre at the origin, passing through the points (a, 0), (-a, 0), (0, b), (0, -b).

O The origin is at the centre of the ellipse. x O x The origin is at the centre of the ellipse. x x x So, we have the parametric equations of an ellipse ( which we met in Cartesian form in Transformations ).

Bilborough College Maths - core 4 parametric equations (Adrian) 24/04/2017 To write the ellipse in Cartesian form we use the same trig identity as we used for the circle. So, for use The equation is usually left in this form.

Pair up cards into graphs and matching Parametric equations Card Match Pair up cards into graphs and matching Parametric equations

There are other parametric equations you might be asked to convert to Cartesian equations. For example, those like the ones in the following exercise. Exercise ( Use a trig identity ) 1. 2. Sketch both curves using either parametric or Cartesian equations. ( Use a graphical calculator if you like ).

1. Solution: Use We usually write this in a form similar to the ellipse: Notice the minus sign. The curve is a hyperbola.

Sketch: or A hyperbola Asymptotes

( Eliminate t by substitution. ) 2. Solution: Subs. in The curve is a rectangular hyperbola.

Sketch: or Asymptotes A rectangular hyperbola.

Plenary Pretty Parametric Equations 1) x=Sin 45t y=sin22.5t 2) x=sin22.5t y=sin11.25t

Bilborough College Maths - core 4 parametric equations (Adrian) 24/04/2017 notes

Bilborough College Maths - core 4 parametric equations (Adrian) 24/04/2017 The Cartesian equation of a curve in a plane is an equation linking x and y. Some of these equations can be written in a way that is easier to differentiate by using 2 equations, one giving x and one giving y, both in terms of a 3rd variable, the parameter. Letters commonly used for parameters are s, t and q. ( q is often used if the parameter is an angle. ) e.gs.

Converting between Cartesian and Parametric forms We use parametric equations because they are simpler, so we only convert to Cartesian if asked to do so ! e.g. 1 Change the following to a Cartesian equation and sketch its graph: Solution: We need to eliminate the parameter t. Substitution is the easiest way. Subst. in

The Cartesian equation is We usually write this as Either, we can sketch using a graphical calculator with and entering the graph in 2 parts. Or, we can notice that the equation is quadratic with x and y swapped over from the more usual form.

e.g. 2 Change the following to a Cartesian equation: Solution: We need to eliminate the parameter q. BUT q appears in 2 forms: as and so, we need a link between these 2 forms. To eliminate q we substitute into the expression.

Multiply by 9: becomes So, N.B. = not We have a circle, centre (0, 0), radius 3.

The following equations give curves you need to recognise: a circle, radius r, centre the origin. a parabola, passing through the origin, with the x-axis an axis of symmetry. an ellipse with centre at the origin, passing through the points (a, 0), (-a, 0), (0, b), (0, -b).