C4 parametric curves_lesson

A2 Mathematics: C4 Core Maths

Curves and Tangents

Parametric Curves

Objectives

We will be able to Plot Graphs defined by parametric equations

– by hand and – by calculator

Use algebra to eliminate the parameter and find the Cartesian equation of the curve.

Find the gradient of the curve for any value of the parameter.

Find the equation of the tangent or normal to the curve at any value of the parameter.

What is a Parametric Graph ?

To plot a graph we could follow a point

– as it crawls – along the curve

especially– If the point obeys a rule

If it gives x and y– In terms of time – Or other parameter

Tracing out a Parametric Graph

http://www.flashandmath.com/mathlets/calc/param2d/param_advanced.html

Tracing out a Parametric Graph

This also shown in your WEC text-book

On page 323

Parametric Curve examples

Parametric Curve examples

Parametric Curve examples

Parametric Equations for a Curve

x=2t, y=15t– 5t²

t

x

y

Plotting x and y via parameters

Curves defined by parametric equations

Parametric Equations for a Curve

x = 3cosθ, y = 3sinθ

t

x

y

Plotting x and y via parameters

Curves defined by parametric equations

Plotting Parametric Curves

Use Sharp EL9900 calculator– Parametric settings on next slide

Use Autograph– Equation entry via x=2t, y=t^2– Separated by a comma

Parametric Settings for EL9900

Parametric Entry for EL9900

Parametric Displays on the EL9900

Cartesian Equation for a Curve

We have x as a function of t or θ

And

y as a function of t or θ

We need to eliminate t or θ

Leaving only x and y.

Methods

1. Eliminate t by substitution and algebra

2. Eliminate θ via trigonometric Identities and algebra

Cartesian Equation – Eliminate t

x = t2, y = t – t2

Cartesian Equation – identities in θ

x = 3cosθ, y = sinθ

Activity step 1

1. Use table of values to plot each curve (and/or use your calculator).

2. Match each parameter formula and its curve with correct curve card.

3. Match each curves with its correct Cartesian equation.

Parametric Equations for a Curve

x = 3cosθ, y = 3sinθ

t

x

y

Cartesian Equation for a Curve

x2 + y2 = 9

Curves defined by parametric equations

Parametric Equations for a Curve

x=2t, y=15t– 5t²

t

x

y

Cartesian Equation for a Curve

4y = 15x– 4.9x2

Curves defined by parametric equations

Parametric Equations for a Curve

x=t²–4, y=t³–4t

t

x

y

Cartesian Equation for a Curve

y = x√(x+4)

y = x(x+4)0.5

Curves defined by parametric equations

Parametric Equations for a Curve

x=sinθ, y=sin2θ

t

x

y

Cartesian Equation for a Curve

y = 2x√(1-x2)

y = 2x(1-x2)0.5

Curves defined by parametric equations

Parametric Equations for a Curve

x=t2, y=t3

t

x

y

Cartesian Equation for a Curve

y=x√x

Curves defined by parametric equations

Parametric Equations for a Curve

x=t, y=1/t

t

x

y

Cartesian Equation for a Curve

y = 1/x

Curves defined by parametric equations

Parametric Equations for a Curve

x = 1+ t, y = 2 - t

t

x

y

Cartesian Equation for a Curve

x + y = 3

Curves defined by parametric equations

Parametric Equations for a Curve

x=(2+3t)/(1+t), y=(3–2t)/(1+t)

t

x

y

Parametric Equations for a Curve

y=13–5x

Curves defined by parametric equations

Stops here !

Extension: Try these Parameters

1. x= t + 1/t, y= t - 1/t

2. x = 3cosθ, y= sinθ

3. Investigate/Create your own

Parametric Equations for a Curve

x=………...., y=…….……..

t

x

y

Curves defined by parametric equations

Tangents to the curve?

How do we find dy/dx ?

How do we find the equation of the tangent at one particular point on the curve

– for example when t=1

Parametric Equations for a Curve

x = 3cosθ, y = 3sinθ

t

x

y

Gradient of Tangents to the Curve

We know (why?) that

Gradient of Tangents to the Curve

x = 3cosθ, y = 3sinθ...so.....

Gradient of Tangents to the Curve

Putting it together.......

How do we find a particular tangent?

Given a particular t value

find x and y, and dy/dx

Now we have the gradient of the tangent and the co-ordinates where it touches the curve

.......so.....

Equation of one Tangent to Circle

Equation of one Tangent to Circle

x = 3cosπ/4, y = 3sin π/4

...so.....

Image of one Tangent to the Curve

Activity step 2

Use x and y parameter functions, to match dy/dx equation one tangent equation

with previous cards

Parametric Equations for a Curve

x=2t, y=15t– 5t²

t

x

y

Gradient of Tangents to the Curve

Image of one Tangent to the Curve

Equation of one Tangent to the Curve

Parametric Equations for a Curve

x=t²–4, y=t³–4t

t

x

y

Gradient of Tangents to the Curve

Image of one Tangent to the Curve

Equation of One Tangent to the Curve

Parametric Equations for a Curve

x=sinθ, y=sin2θ

t

x

y

Gradient of Tangents to the Curve

Image of Tangent to Curve

Equation of One Tangent to the Curve

Parametric Equations for a Curve

x=t2, y=t3

t

x

y

Gradient of Tangents to the Curve

Image of Tangent to Curve

Equation of one Tangent to the Curve

Parametric Equations for a Curve

x=t, y=1/t

t

x

y

Gradient of Tangents to the Curve

Image of Tangent to Curve

Equation of one Tangent to the Curve

Parametric Equations for a Curve

x = 1+ t, y = 2 - t

t

x

y

Gradient of Tangents to the Curve

Image of Tangent to Curve

Equation of one Tangent to the Curve

Parametric Equations for a Curve

x=(2+3t)/(1+t), y=(3–2t)/(1+t)

t

x

y

Gradient of Tangents to the Curve

Image of Tangent to Curve

Equation of one Tangent to the Curve

Parametric Equations for a Curve

x=………...., y=…….……..

t

x

y

Curves defined by parametric equations

Gradient of Tangents to the Curve

Equation of one Tangent to the Curve

Image of Tangent to Curve