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