What is the purpose of parametric equations? (1 Viewer)

[Masaken]

Genuine question because the books I'm using + the internet are a bit confusing to me.

I get parametric equations are an alternate way to the Cartesian form to represent functions (kinda?), but what are they used for + what is their purpose? Do they have any applications within the course, and if so, what are they? I've only gotten so far in this topic where I know how to convert Cartesian equations to parametric equations and vice versa, and started learning how to sketch them but apart from that I have no idea what the point of them are.

 

[notme123]

they were more useful in the old syllabus where parabolas would be represented as parametric equations in terms of t, or the gradient of the tangent at that point. but theyre still useful. for example representing circles in terms of the angle between the x-axis and the line from the origin to a point on the circle. they usually test them in one way or another so learn how to represent, draw and interpret circles in paametric form and try to find harder ones that use trig identities. if youre looking for a hsc example look at q8 of last years mc.

 

[ExtremelyBoredUser]

Genuine question because the books I'm using + the internet are a bit confusing to me.

I get parametric equations are an alternate way to the Cartesian form to represent functions (kinda?), but what are they used for + what is their purpose? Do they have any applications within the course, and if so, what are they? I've only gotten so far in this topic where I know how to convert Cartesian equations to parametric equations and vice versa, and started learning how to sketch them but apart from that I have no idea what the point of them are.

you will use them heavily in Projectile Motion (Ext 1 & 2) as well as 3D Vectors (Ext 2) but its like the really basic part of just substituting variables... so might be common sense by then..?

e.g




(not the best examples but I just found them straight off the 2020 HSC Ext 2 Paper since I had that tab open)

the whole part of eliminating variables will be important but I doubt parametric equations as a whole is as important as before with locus/parabola questions (which is old syllabus). I think the only important part for parametric equations is being able to convert them into cartesian form and perhaps converting cartesian form into parametric form. I'm sure old textbooks have those parabola type qs that notme referred to but I wouldn't bother since from what I've seen, its not at all covered/referenced.

 

[Masaken]

Thanks for that!

Oh, and Locus is that all points thing, right? (I remember it being taught in a single maths lesson in Year 7, then never touched upon ever again)

 

[ExtremelyBoredUser]

Thanks for that!

Oh, and Locus is that all points thing, right? (I remember it being taught in a single maths lesson in Year 7, then never touched upon ever again)

Yeah pretty much. Set of points which satisfy an equation/relationship between variables. I've never seen the term doing syllabus content besides for Complex Numbers which it is heavily used there to refer to circles, hyperbolas, ellipses etc. on the complex plane. I rmb doing such type of qs relating to locus of a parabola in year 10 but I forgot...

 

[Eagle Mum]

The concept of a Cartesian plane incorporates the notion of an independent variable x and a dependent variable y that is a function of x, but, as per extremelyboreduser’s example of projectile motion, there may be a relationship between x and y, but they are dependent on other variable(s) rather than y being dependent on x (eg. vertical and horizontal components of motion are dependent on time and angle of trajectory).

 

[Eagle Mum]

Yeah pretty much. Set of points which satisfy an equation/relationship between variables. I've never seen the term doing syllabus content besides for Complex Numbers which it is heavily used there to refer to circles, hyperbolas, ellipses etc. on the complex plane. I rmb doing such type of qs relating to locus of a parabola in year 10 but I forgot...

Apart from the parabola, all the other conic sections (circles, ellipse, hyperbolas) have two y values for each x, so by definition, their Cartesian equations aren’t functions, whereas the dual parametric equations (for each single Cartesian equation) are functions.

 

[Trebla]

Parametric equations can be used to describe curves which may not necessarily be easily expressed in Cartesian equations.

This is particularly the case once you start going beyond two dimensions, For example, the equation of a helix:
x = cos t
y = sin t
z = t

The x-y parameters are that of a circle and the z parameter shows this circle increasing in elevation as t increases.