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# C4 parametric curves_lesson

## on Apr 04, 2011

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A2-level maths UK

A2-level maths UK

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## C4 parametric curves_lessonPresentation Transcript

• 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
• 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
• Plotting x and y via parameters
• Curves defined by parametric equations
• Parametric Equations for a Curve
x = 3cosθ, y = 3sinθ
• 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
yas a function of t or θ
We need to eliminate t or θ
Leaving only x and y.
Methods
Eliminate t by substitution and algebra
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
Use table of values to plot each curve (and/or use your calculator).
Match each parameter formula and its curve with correct curve card.
Match each curves with its correct Cartesian equation.
• Parametric Equations for a Curve
x = 3cosθ, y = 3sinθ
• Cartesian Equation for a Curve
x2 + y2 = 9
• Curves defined by parametric equations
• Parametric Equations for a Curve
x=2t, y=15t– 5t²
• 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
• 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
• 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
• Cartesian Equation for a Curve
y=x√x
• Curves defined by parametric equations
• Parametric Equations for a Curve
x=t, y=1/t
• Cartesian Equation for a Curve
y = 1/x
• Curves defined by parametric equations
• Parametric Equations for a Curve
x = 1+ t, y = 2 - t
• 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)
• Parametric Equations for a Curve
y=13–5x
• Curves defined by parametric equations
Stops here !
• Extension: Try these Parameters
x= t + 1/t, y= t - 1/t
x = 3cosθ, y= sinθ
• Parametric Equations for a Curve
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
• 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
• 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
• 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
• 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
• 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
• 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
• 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)
• 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=…….……..
• Curves defined by parametric equations
• Gradient of Tangents to the Curve
• Equation of one Tangent to the Curve
• Image of Tangent to Curve