3.3.1

Parametric Equations

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What are Parametric Equations?

Parametric equations define the coordinates of a graph as functions of a third variable, tt. Each coordinate of the graph is given by the point (x=f(t),y=g(t))(x=f(t),y=g(t)).

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Definition

  • We can define a curve using the equations:
    • x=f(t)x = f(t) and y=g(t)y = g(t)
  • These are called parametric equations.
  • Parametric equations define a point on the curve in terms of a parameter tt.
  • You must specify the domain of tt that the curve exists for when defining a curve parametrically.
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Example

  • A curve has parametric equations:
    • x=t1,  y=2t+4,  3t2x = t-1,\; y = 2t+4, \;-3\leq t \leq 2
  • By substituting in values of tt in the domain given, we can find the coordinates of points on the curve.
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Finding points on the curve

  • For example, when t=1t=-1:
    • x=11=2x = -1-1 = -2
    • y=(2×1)+4=2y=(2\times -1) +4=2.
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Conversion

  • If a curve is only in terms of the variables xx and yy, we call this a Cartesian equation.
  • We can convert parametric equations into a Cartesian equation by rearranging for tt and substituting it in.
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Example

  • A curve has parametric equations:
    • x=t1,  y=2t+4,  3t2x = t-1,\; y = 2t+4, \;-3\leq t \leq 2
  • Rearranging the first equation, we get:
    • x=t1t=x+1x = t-1 \rightarrow t = x+1
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Substitute in tt

  • Substituting this into the equation for yy:
    • y=2(x+1)+4=2x+6y = 2(x+1)+4 = 2x+6
  • That means the Cartesian equation for this curve is y=2x+6y = 2x+6
  • However, we can't finish there, as the curve was specified for the region 3t2-3\leq t\leq2.
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Find the domain

  • We need to substitute in for tt into the inequality that describes its domain:
    • 3x+12-3\leq x+1\leq2
  • Subtracting one from each side gives:
    • 4x1-4\leq x\leq 1
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Cartesian equation

  • So the curve with parametric equations:
    • x=t1,  y=2t+4,  3t2x = t-1,\; y = 2t+4, \;-3\leq t \leq 2
  • Is the same as the Cartesian equation:
    • y=2x+6,4x1y = 2x+6, -4\leq x\leq1

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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