3.3.1
Parametric Equations
What are Parametric Equations?
What are Parametric Equations?
Parametric equations define the coordinates of a graph as functions of a third variable, . Each coordinate of the graph is given by the point .
Definition
Definition
- We can define a curve using the equations:
- and
- These are called parametric equations.
- Parametric equations define a point on the curve in terms of a parameter .
- You must specify the domain of that the curve exists for when defining a curve parametrically.
Example
Example
- A curve has parametric equations:
- By substituting in values of in the domain given, we can find the coordinates of points on the curve.
Finding points on the curve
Finding points on the curve
- For example, when :
- .
Conversion
Conversion
- If a curve is only in terms of the variables and , we call this a Cartesian equation.
- We can convert parametric equations into a Cartesian equation by rearranging for and substituting it in.
Example
Example
- A curve has parametric equations:
- Rearranging the first equation, we get:
Substitute in
Substitute in
- Substituting this into the equation for :
- That means the Cartesian equation for this curve is
- However, we can't finish there, as the curve was specified for the region .
Find the domain
Find the domain
- We need to substitute in for into the inequality that describes its domain:
- Subtracting one from each side gives:
Cartesian equation
Cartesian equation
- So the curve with parametric equations:
- Is the same as the Cartesian equation:
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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