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10.7 Parametric Equations: Graphs

3 min read•june 24, 2024

offer a unique way to describe curves using an independent . They allow us to express complex shapes and motions that might be challenging to represent with standard functions. By defining x and y coordinates separately, we gain flexibility in modeling various phenomena.

From simple lines to intricate cycloids, parametric equations open up a world of mathematical possibilities. They find practical applications in physics, engineering, and economics, helping us understand everything from planetary orbits to economic cycles. This powerful tool expands our ability to analyze and represent real-world situations mathematically.

Parametric Equations and Their Graphs

Plotting parametric equation points

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  • Set of equations express coordinates of points on a plane curve using independent parameter tt
    • defined by function x(t)x(t)
    • defined by function y(t)y(t)
  • To plot points and graph:
    1. Choose range of tt values
    2. Substitute each tt value into equations to find corresponding xx and yy coordinates
    3. Plot resulting points (x,y)(x, y) on coordinate plane (in )
    4. Connect points smoothly to create graph of parametric curve
  • Examples:
    • Line: x(t)=2t+1x(t) = 2t + 1, y(t)=3t−2y(t) = 3t - 2
    • Circle: x(t)=3cos⁡(t)x(t) = 3\cos(t), y(t)=3sin⁡(t)y(t) = 3\sin(t)

Shape analysis of parametric curves

  • Analyze shape and by:
    • Examining and y-coordinate functions separately
      • Determine domain and range of each function
      • Identify symmetries, periodicities, or asymptotes
    • Considering relationship between x-coordinate and y-coordinate functions
      • Look for patterns or similarities in formulas
      • Determine if curve is closed (ends where it starts) or open
    • Investigating behavior of curve as tt approaches infinity or negative infinity
      • Determine if curve has horizontal or vertical asymptotes
      • Identify direction of curve as tt increases or decreases
  • Common types of :
    • Lines: x(t)=at+bx(t) = at + b, y(t)=ct+dy(t) = ct + d (aa, bb, cc, dd are constants)
    • Circles: x(t)=rcos⁡(t)x(t) = r \cos(t), y(t)=rsin⁡(t)y(t) = r \sin(t) (rr is radius)
    • Ellipses: x(t)=acos⁡(t)x(t) = a \cos(t), y(t)=bsin⁡(t)y(t) = b \sin(t) (aa and bb are semi-major and semi-minor axes)
    • Cycloids: x(t)=r(t−sin⁡(t))x(t) = r(t - \sin(t)), y(t)=r(1−cos⁡(t))y(t) = r(1 - \cos(t)) (rr is radius of generating circle)

Real-world applications of parametric equations

  • Model various situations, such as:
    • Projectile motion: path of object launched at angle to horizontal
      • x-coordinate represents horizontal distance traveled
      • y-coordinate represents vertical height of object
      • Example: x(t)=v0cos⁡(θ)tx(t) = v_0 \cos(\theta) t, y(t)=v0sin⁡(θ)t−12gt2y(t) = v_0 \sin(\theta) t - \frac{1}{2}gt^2 (v0v_0 is initial velocity, θ\theta is launch angle, gg is acceleration due to gravity)
    • Planetary motion: path of planet orbiting sun
      • x-coordinate and y-coordinate functions describe planet's position in orbital plane
      • Kepler's laws of planetary motion can be expressed using parametric equations
    • Cyclical phenomena: situations that repeat periodically (tides, seasons, economic cycles)
      • Parameter tt represents time
      • x-coordinate and y-coordinate functions describe how phenomenon varies over time
  • Examples:
    • Ferris wheel: x(t)=20cos⁡(t)x(t) = 20\cos(t), y(t)=20sin⁡(t)+30y(t) = 20\sin(t) + 30 (20 is radius, 30 is height of center)
    • Spiral: x(t)=tcos⁡(t)x(t) = t\cos(t), y(t)=tsin⁡(t)y(t) = t\sin(t)

Alternative Coordinate Systems and Functions

  • : an alternative to Cartesian coordinates for describing points in a plane
  • : functions that output vectors, often used to describe curves in space
  • : a graphical representation of a system's behavior, particularly useful in dynamical systems

Key Terms to Review (28)

Cardioid: A cardioid is a plane curve that resembles a heart shape. It is a type of cycloid curve that is generated by a point on the circumference of a circle as it rolls along a straight line. The cardioid has a distinctive heart-like appearance and is often used in various mathematical and scientific applications.
Cartesian Coordinates: Cartesian coordinates are a system used to locate points in a two-dimensional or three-dimensional space. This system uses a pair or trio of numerical coordinates to uniquely identify the position of a point relative to a fixed reference frame.
Complement of an event: The complement of an event is the set of all outcomes in a sample space that are not included in the event. It is denoted as $E'$ or $E^c$ where $E$ is the event.
Cosine Function: The cosine function is a periodic function that describes the x-coordinate of a point moving in a circular path. It is one of the fundamental trigonometric functions, along with the sine function, and is widely used in various mathematical and scientific applications.
Cycloid: A cycloid is a curve traced by a point on the circumference of a circle as it rolls along a straight line. It is a roulette curve, meaning it is generated by the motion of one curve rolling on another.
Elimination Method: The elimination method, also known as the method of elimination, is a technique used to solve systems of linear equations by systematically eliminating variables to find the unique solution. This method is applicable in the context of various topics, including parametric equations, systems of linear equations in two and three variables, and systems of nonlinear equations and inequalities.
Event: An event is a specific outcome or set of outcomes of a random experiment. It is a subset of the sample space.
Experiment: An experiment is a process or action that generates observable outcomes. It is used to measure the likelihood of various results in a probabilistic context.
Implicit Form: Implicit form refers to the representation of a function or equation where the relationship between the variables is not explicitly stated, but rather defined through an equation that involves both variables. This form is commonly used in the context of parametric equations and their graphs.
Intersection: The intersection of two events in probability is the set of outcomes that are common to both events. It is denoted by $A \cap B$.
Lemniscate: A lemniscate is a plane curve that resembles a figure eight. It is a closed curve that has a distinctive shape with two loops that intersect at a central point. The lemniscate is an important concept in both polar coordinates and parametric equations.
Mutually exclusive events: Mutually exclusive events are events that cannot occur at the same time. If one event happens, the other cannot.
Orientation: Orientation refers to the position or arrangement of an object or entity in relation to a frame of reference, such as a coordinate system or a particular direction. It is a fundamental concept in the context of parametric equations and their graphical representations.
Outcomes: Outcomes are the possible results that can occur from a probabilistic experiment. Each outcome is a single realization of what could happen when the experiment is conducted.
Parameter: A parameter is a variable that serves as an input to a function or equation, allowing it to be adjusted or changed to produce different results. It is a quantity that defines the characteristics or behavior of a mathematical model or system.
Parametric Curves: Parametric curves are a way of representing a curve in the coordinate plane using a pair of parametric equations. These equations define the x and y coordinates of the curve as functions of a third variable, known as the parameter.
Parametric Equations: Parametric equations are a way of representing the coordinates of a point as functions of a parameter, typically denoted by the variable 't'. This allows for the description of curves and shapes that cannot be easily represented using traditional Cartesian coordinates.
Parametric Form: Parametric form is a way of representing a curve or function by expressing the coordinates of points on the curve as functions of a third variable, called a parameter. This allows for a more flexible and versatile description of the curve compared to using a single equation in terms of the x and y variables.
Phase Plane: The phase plane is a graphical representation of the relationship between the position and velocity of a dynamic system. It is a powerful tool used in the analysis of parametric equations, which describe the motion of an object in a two-dimensional plane as a function of a parameter, such as time.
Polar Coordinates: Polar coordinates are a system of representing points in a plane using the distance from a fixed point (the pole) and the angle from a fixed direction (the polar axis). This system provides an alternative to the Cartesian coordinate system and is particularly useful for describing circular and periodic phenomena.
Probability model: A probability model is a mathematical representation of a random phenomenon. It consists of a sample space, events within the sample space, and probabilities associated with each event.
Sine Function: The sine function is a periodic function that describes the y-coordinate of a point moving around the unit circle. It is one of the fundamental trigonometric functions and is widely used in various fields, including mathematics, physics, and engineering.
Traced Curve: A traced curve, in the context of parametric equations, is a path or trajectory that is generated by plotting the coordinates defined by a set of parametric equations. These equations describe the position of a point as a function of a variable, typically denoted as the parameter, which can represent time or any other independent variable.
Union of two events: The union of two events in probability is the event that either one or both of the events occur. It is denoted as $A \cup B$.
Vector-Valued Functions: A vector-valued function is a function that assigns a vector, rather than a scalar, to each input value. These functions are often used to describe the motion of an object in two or three-dimensional space, where the vector represents the position, velocity, or acceleration of the object at a given time.
X-coordinate: The x-coordinate is the first value in an ordered pair $(x, y)$ representing a point's horizontal position on the Cartesian plane. It indicates how far left or right the point is from the origin (0, 0).
X-Coordinate: The x-coordinate is the horizontal position of a point on a coordinate plane. It represents the distance from the origin (0,0) to the point along the horizontal x-axis. The x-coordinate is a crucial component in understanding and working with various mathematical concepts, including coordinate systems, graphs, unit circles, and systems of linear equations.
Y-coordinate: The y-coordinate is the vertical position of a point on a coordinate plane, measured as the distance from the x-axis. It represents the up-down position of a point and is used to describe the location of objects or data points within a two-dimensional coordinate system.
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10.8 Vectors