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41: Parametric Equations “Teach A Level Maths” Vol. 2: A2 Core Modules

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Parametric Equations 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 3 rd variable, the parameter. Letters commonly used for parameters are s, t and . ( is often used if the parameter is an angle. ) e.gs.

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Parametric Equations 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. We substitute for t from the easier equation: Subst. in

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Parametric Equations 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.

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Parametric Equations The sketch is The curve is called a parabola. Also, the parametric equations show that as t increases, x increases faster than y.

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Parametric Equations e.g. 2 Change the following to a Cartesian equation: Solution: We need to eliminate the parameter . BUT appears in 2 forms: as and so, we need a link between these 2 forms. Which trig identity links and ? ANS: To eliminate we substitute into this expression.

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

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Parametric Equations Since we recognise the circle in Cartesian form, it’s easy to sketch. However, if we couldn’t eliminate the parameter or didn’t recognise the curve having done it, we can sketch from the parametric form.

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Parametric 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 : and ( It doesn’t matter that we don’t know which angle is measuring. ) For both and the min is 1 and the max is 1, so e.g. Sketch the curve with equations It’s also easy to get the other coordinate at each of these 4 key values e.g.

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Parametric Equations and We could draw up a table of values finding x and y for values of but this is usually very inefficient. Try to just pick out significant features. x x x x

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Parametric Equations and x This tells us what happens to x and y. Think what happens to and as increases from 0 to. We could draw up a table of values finding x and y for values of but this is usually very inefficient. Try to just pick out significant features. x x x x

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Parametric Equations and x Symmetry now completes the diagram. This tells us what happens to x and y. Think what happens to and as increases from 0 to. x x x x

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Parametric Equations and Symmetry now completes the diagram. x x x x

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Parametric Equations and Symmetry now completes the diagram. x x x x

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

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Parametric Equations You can use a graphical calculator to sketch curves given in parametric form. However, you will have to use the setup menu before you enter the equations. You will also have to be careful about the range of values of the parameter and of x and y. If you don’t get the right scales you may not see the whole graph or the graph can be distorted and, for example, a circle can look like an ellipse. By the time you’ve fiddled around it may have been better to sketch without the calculator!

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Parametric Equations 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 ).

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Parametric Equations 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.

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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 ).

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

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Parametric Equations Sketch: or A hyperbola Asymptotes

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

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Parametric Equations Sketch: or A rectangular hyperbola. Asymptotes

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Parametric Equations

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The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

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Parametric Equations 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 3 rd variable, the parameter. Letters commonly used for parameters are s, t and . ( is often used if the parameter is an angle. ) e.gs.

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Parametric Equations 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

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Parametric Equations 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.

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

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

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Parametric Equations 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 ).

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