Glossary

12.3 Eliminating the Parameter

2 min read•july 25, 2024

are a cool way to describe curves using a parameter like time or angle. They're super useful for modeling real-world situations, from projectile motion to planetary orbits.

Converting between parametric and Cartesian forms is key. It helps identify conic sections and their features. Mastering these conversions opens up new ways to analyze and solve problems in geometry and physics.

Parametric Equations and Cartesian Form

Elimination of parametric equations

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  • Parametric equations express x and y coordinates in terms of a parameter t x=f(t)x = f(t), y=g(t)y = g(t) (time, angle)
  • :
    1. Solve one equation for t in terms of x or y
    2. Substitute t expression into other equation
    3. Simplify resulting equation
  • Common techniques:
    • Direct replaces t with equivalent expression
    • rearranges equations
    • used for sine and cosine equations (sin⁡2t+cos⁡2t=1\sin^2t + \cos^2t = 1)

Identification of conic sections

  • Four types: , , ,
  • Analyze eliminated :
    • Circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2
    • Ellipse: (x−h)2a2+(y−k)2b2=1\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
    • Parabola: (y−k)2=4p(x−h)(y - k)^2 = 4p(x - h) or (x−h)2=4p(y−k)(x - h)^2 = 4p(y - k)
    • Hyperbola: (x−h)2a2−(y−k)2b2=1\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
  • Special cases include degenerate conics (point, line, pair of lines) and rotated conics

Equations from parametric form

  • Eliminate parameter to obtain Cartesian equation
  • Identify resulting equation type
  • Determine key features:
    • (h, k)
    • define shape's extent
    • influence curve's shape
    • measures deviation from circular form

Conversion to parametric form

  • Reverse process of parameter elimination
  • General strategies:
    • Use trigonometric functions for circles and ellipses
    • Represent one variable in terms of other using parameter t
  • Specific techniques:
    • Circle: x=rcos⁡(t)x = r\cos(t), y=rsin⁡(t)y = r\sin(t)
    • Ellipse: x=acos⁡(t)x = a\cos(t), y=bsin⁡(t)y = b\sin(t)
    • Parabola: [x = at^2](https://www.fiveableKeyTerm:x_=_at^2), [y = 2at](https://www.fiveableKeyTerm:y_=_2at) (vertical) or [x = 2at](https://www.fiveableKeyTerm:x_=_2at), [y = at^2](https://www.fiveableKeyTerm:y_=_at^2) (horizontal)
    • Hyperbola: x=asec⁡(t)x = a\sec(t), y=btan⁡(t)y = b\tan(t) or x=acosh⁡(t)x = a\cosh(t), y=bsinh⁡(t)y = b\sinh(t)
  • Verify conversion by eliminating parameter to recover original Cartesian equation

Key Terms to Review (28)

Algebraic manipulation: Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions to make calculations easier or to solve equations. This technique is vital in mathematics as it allows for the application of fundamental principles to derive new forms of expressions, solve for unknowns, and apply identities effectively. Mastery of algebraic manipulation is essential for manipulating trigonometric identities and eliminating parameters in various mathematical contexts.
Angle parameter: An angle parameter is a variable that represents an angle in a parametric equation, allowing for the representation of curves and shapes in a coordinate system. By using angle parameters, equations can describe circular motion or periodic functions, making them essential for transforming complex relationships into simpler forms. This concept is critical when transitioning between parametric and Cartesian forms of equations.
Cartesian Equation: A Cartesian equation is a mathematical expression that defines a relationship between variables in a coordinate system, typically using x and y coordinates. This type of equation can represent various geometric shapes such as lines, curves, and surfaces in a two-dimensional or three-dimensional space, allowing for the analysis of their properties and behaviors. By transforming parametric equations into Cartesian equations, one can eliminate the parameter and describe the relationship purely in terms of the coordinates.
Center: In mathematics, particularly in the study of parametric equations and curves, the center refers to a specific point that serves as a reference for the geometric properties of the shape being described. This term is crucial when eliminating the parameter in parametric equations, as it often helps to identify the position and symmetry of a curve, guiding the transformation into Cartesian form.
Circle: A circle is a two-dimensional shape defined as the set of all points in a plane that are equidistant from a fixed point known as the center. This geometric figure has key properties, including its radius, which is the distance from the center to any point on the circle, and its diameter, which is twice the radius. The concept of a circle plays an essential role in various mathematical contexts, especially when dealing with parametric equations and understanding the elimination of parameters.
Domain of t: The domain of t refers to the set of all possible values that the parameter t can take in a parametric equation. Understanding the domain is crucial for interpreting the behavior and characteristics of the resulting curve or geometric figure formed by the equations. It plays a significant role in eliminating the parameter, as it helps to determine valid intervals for the variable that describe the shape of the graph without explicitly involving t.
Eccentricity: Eccentricity is a measure of how much a conic section deviates from being circular, represented by the ratio of the distance between the foci and the length of the major axis. It provides important insights into the shape of conic sections like ellipses and hyperbolas, where lower values indicate rounder shapes and higher values indicate more elongated forms. Understanding eccentricity is crucial when eliminating parameters in equations to find standard forms of these shapes.
Elimination process: The elimination process is a mathematical technique used to eliminate a parameter from a set of parametric equations, resulting in a single equation that describes the relationship between the variables. This method simplifies complex relationships and allows for easier analysis of curves and shapes represented in the coordinate system. By substituting one equation into another, the elimination process ultimately transforms parametric equations into Cartesian coordinates.
Ellipse: An ellipse is a closed curve that results from the intersection of a cone with a plane that is tilted relative to the axis of the cone. This shape is characterized by its two focal points, and the sum of the distances from any point on the ellipse to these foci is constant. In the context of parametric equations, an ellipse can be represented using sinusoidal functions, allowing for a clear visualization of its properties. When eliminating the parameter, the standard equation of the ellipse can be derived, providing further insight into its geometric characteristics.
Foci: In the context of conic sections, foci are the fixed points that help define the shape and properties of a conic, such as ellipses and hyperbolas. The distance from any point on the conic to each focus plays a crucial role in determining the conic's specific characteristics, such as its eccentricity and overall shape.
Hyperbola: A hyperbola is a type of conic section that occurs when a plane intersects both halves of a double cone. It consists of two separate curves called branches, which are mirror images of each other and open either horizontally or vertically. This unique structure is defined by its relationship to its foci and asymptotes, making it an essential concept in the study of parametric equations and the process of eliminating parameters.
Parabola: A parabola is a U-shaped curve that is defined as the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. Parabolas are significant in various mathematical contexts, particularly in understanding the behavior of quadratic functions and their geometric representations. They can be expressed in parametric equations, which describe the coordinates of points along the curve, and can also be analyzed by eliminating the parameter to find standard forms.
Parametric equations: Parametric equations are a set of equations that express the coordinates of points in a geometric figure as functions of a variable, usually denoted as 't'. This approach allows for the representation of curves and shapes that might be difficult to describe using standard Cartesian coordinates. By introducing a parameter, these equations can effectively capture motion, orientation, and other dynamic properties.
Pythagorean Identity: The Pythagorean Identity is a fundamental relationship in trigonometry that expresses the square of the sine function plus the square of the cosine function as equal to one, represented as $$ ext{sin}^2(x) + ext{cos}^2(x) = 1$$. This identity is crucial in connecting the concepts of angle measures and the unit circle, forming a basis for deriving other identities and solving various trigonometric problems.
Range of y: The range of y refers to the set of all possible output values (y-values) that a function or relation can produce, given its input values (x-values). Understanding the range is crucial in determining how the output behaves as the input varies, especially when eliminating parameters to express relationships in a more straightforward manner. It helps in visualizing and interpreting the behavior of functions graphically and algebraically.
Substitution: Substitution refers to the process of replacing a variable or expression with another equivalent variable or expression in mathematical equations. This technique is particularly useful in solving trigonometric equations and simplifying expressions, as it allows for easier manipulation and understanding of complex relationships. By substituting values or identities, one can derive solutions that are often more manageable and reveal deeper insights into the behavior of trigonometric functions.
Time parameter: A time parameter is a variable that represents time within a mathematical model or equation, often used in the context of parametric equations to describe the relationship between variables. By introducing a time parameter, it allows for the representation of motion or change over time, enabling the analysis of dynamic systems and behaviors. This concept is particularly useful when eliminating parameters to express a relationship purely in terms of other variables.
Trigonometric Identities: Trigonometric identities are equations that involve trigonometric functions and hold true for all values of the variables involved, where these functions are defined. They are fundamental in simplifying trigonometric expressions, solving equations, and proving other mathematical statements. Mastery of these identities allows for deeper understanding and manipulation of relationships between angles and sides in various mathematical contexts.
Vertices: Vertices are the distinct points where two or more edges meet in geometric figures. In the context of eliminating parameters, vertices play a crucial role as they can represent critical points in parametric equations, allowing us to convert these equations into Cartesian forms. Understanding how vertices relate to the shapes created by these equations helps in visualizing and interpreting geometric transformations.
X = 2at: The equation x = 2at relates the horizontal displacement 'x' of a projectile to the time 't' it spends in the air and the acceleration 'a' due to gravity. This equation is crucial in understanding how objects move under constant acceleration, especially when discussing projectile motion. It showcases the linear relationship between displacement and time while also emphasizing the role of acceleration, making it essential for analyzing trajectories.
X = a cos(t): The equation x = a cos(t) describes the x-coordinate of a point on a circle of radius 'a' as a function of the angle 't'. This relationship is essential for understanding parametric equations, where points are expressed in terms of parameters like angles, and helps illustrate how circular motion can be modeled mathematically. It serves as a fundamental aspect of converting between parametric and Cartesian forms in trigonometric contexts.
X = a sec(t): The equation x = a sec(t) represents a relationship between the variable x and the secant function of the angle t, scaled by a constant a. This term is essential in parametric equations where x is expressed in terms of an angle and often relates to the geometry of curves, such as hyperbolas, that can be described using trigonometric functions. Understanding this equation allows for the conversion of parametric forms to rectangular coordinates, aiding in visualizing and analyzing the behavior of curves.
X = at^2: The equation x = at^2 represents a quadratic relationship between the variable x and time t, where 'a' is a constant that determines the curvature of the graph. This equation describes a parabolic trajectory, which is commonly found in motion under constant acceleration. In this context, it illustrates how the position changes over time as an object accelerates, connecting to the broader concept of parametric equations where both x and y coordinates depend on an external parameter, typically time.
X = r cos(t): The equation x = r cos(t) represents the relationship between Cartesian and polar coordinates, where 'x' is the horizontal coordinate, 'r' is the distance from the origin to a point in the plane, and 't' (theta) is the angle measured from the positive x-axis. This formula is essential for converting polar coordinates to Cartesian coordinates, allowing for the analysis of curves described in polar form.
Y = 2at: The equation y = 2at describes the relationship between the variable 'y', a constant 'a', and the parameter 't' in the context of parametric equations. This equation can arise when eliminating the parameter from a parametric representation of a curve, particularly in cases involving projectile motion or quadratic relationships. It showcases how a linear relation can be derived from a parameterized form, illustrating a key principle in calculus and analytical geometry.
Y = at^2: The equation y = at^2 represents a quadratic relationship between the variable 'y' and time 't', where 'a' is a constant that affects the curvature of the graph. This equation is significant because it models motion in physics, such as the trajectory of an object under uniform acceleration. Understanding this relationship helps in analyzing how changes in time impact the value of 'y', making it a key component in various applications, including kinematics and parametric equations.
Y = b sin(t): The equation y = b sin(t) represents a sinusoidal function where 'b' indicates the amplitude of the sine wave. This term is critical for understanding how periodic functions behave, particularly in relation to motion and oscillations. The variable 't' often denotes time, leading to applications in modeling wave patterns and harmonic motion, which are essential in various fields such as physics and engineering.
Y = r sin(t): The equation y = r sin(t) represents the vertical coordinate of a point in polar coordinates, where 'r' is the distance from the origin and 't' is the angle measured from the positive x-axis. This relationship connects polar coordinates to Cartesian coordinates, allowing for the analysis and visualization of curves and shapes in a two-dimensional plane. Understanding this equation is crucial for converting between coordinate systems and can reveal insights about periodic functions and their properties.
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