offer a powerful way to describe curves using two equations and a parameter. They're especially handy for circles, ellipses, and projectile motion. By plotting points or converting to Cartesian form, we can visualize and analyze these curves.
Time-dependent representations make parametric equations crucial in physics and engineering. They allow us to model motion and trajectories, opening up a world of practical applications. Understanding parametric equations enhances our ability to describe and analyze complex curves and movements.
Parametric Equations and Graphs
Plotting parametric curve points
Parametric equations define plane curves using two equations and a parameter t
x-coordinate equation x=f(t) and y-coordinate equation y=g(t)
Substitute t values into both equations to find corresponding (x,y) coordinates (cartesian coordinates)
x=2t and y=t2, when t=1, the point is (2,1) and when t=−2, the point is (−4,4)
Plot the points and connect them to create the parametric curve graph
Interpreting common parametric graphs
Circles represented parametrically using trigonometric functions
x=rcost and y=rsint, r is the radius and 0≤t≤2π (r=3 creates a circle with radius 3)
Ellipses represented parametrically using trigonometric functions
x=acost and y=bsint, a and b are the semi-major and semi-minor axes, and 0≤t≤2π (a=4 and b=2 creates an with a horizontal major axis)
Projectile motion modeled using parametric equations
Horizontal motion x=v0cosθ⋅t, v0 is the initial velocity and θ is the launch angle (v0=50 m/s and θ=60°)
Vertical motion y=v0sinθ⋅t−21gt2, g is the acceleration due to (g=9.8 m/s²)
Sketch graphs by identifying key points and features
Circles and ellipses: locate center and vertices
Projectile motion: find maximum height and range
Conversion of parametric to Cartesian equations
Convert parametric to Cartesian form by eliminating parameter t
Solve one equation for t in terms of x or y
Substitute the expression for t into the other equation
Simplify the resulting Cartesian equation
x=2t and y=t2, solve for t=2x and substitute into y=(2x)2 to get y=4x2
Convert Cartesian to by introducing parameter t and expressing x and y in terms of t
y=x2 can be parameterized as x=t and y=t2 or x=cost and y=cos2t
Curves may have multiple valid parametrizations
Circle with radius 1: x=cost, y=sint, or x=sint, y=cost
Time-dependent representations
Parametric equations often represent time-dependent equations in physics and engineering
Vector-valued functions can be used to describe parametric curves in higher dimensions
Parametric curves are useful for modeling motion and trajectories over time
Key Terms to Review (7)
Ellipse: An ellipse is a set of all points in a plane such that the sum of their distances from two fixed points (foci) is constant. It has an oval shape and can be represented by its standard equation.
Gravity: Gravity is the force that attracts objects with mass toward each other. It plays a crucial role in determining the trajectories of objects influenced by parametric equations.
Parabola: A parabola is a symmetric curve formed by the set of all points in a plane that are equidistant from a given point called the focus and a given line called the directrix. It is commonly represented by the quadratic function $y = ax^2 + bx + c$.
Parametric equations: Parametric equations represent a set of related quantities as explicit functions of one or more independent variables called parameters. They are often used to describe the motion of objects and to graph complex curves.
Parametric form: Parametric form describes a set of equations where the coordinates are expressed as functions of one or more independent parameters. Commonly used to represent curves and surfaces in mathematics, parametric equations provide a convenient way to describe complex geometrical shapes.
Pythagorean Theorem: The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it is expressed as $a^2 + b^2 = c^2$ where $c$ is the hypotenuse.
Rectangular form: Rectangular form of a complex number is expressed as $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit. It represents the complex number in terms of its horizontal (real) and vertical (imaginary) components.