Parametric Functions
Definition
Parametric functions are a way to represent curves or graphs using two separate equations, one for the x-coordinate and one for the y-coordinate. These equations are typically in terms of a third variable called the parameter.
Analogy
Think of parametric functions as a pair of synchronized dancers on a stage. Each dancer moves independently, following their own set of instructions (equations), but together they create a beautiful choreography (curve).
Related terms
Cartesian Coordinates: A system that uses perpendicular axes to locate points on a plane. It consists of an x-axis and a y-axis.
Parameter: The independent variable used in parametric equations to determine the values of the x and y coordinates.
Graphing Calculator: A tool that can be used to visualize and analyze parametric functions by plotting points based on given equations.
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Study guides (2)
Practice Questions (10)
- Greg is bowling with his friends and rolls the ball at time t = 0. Consider the center axis of the lane to correspond to line x = 0 and the pin deck to be at the line y = 15. The gutters correspond to the lines x = -4 and x = 4. If a ball falls into a gutter before hitting any pins, Greg’s score is 0. The motion of the ball can be described by the parametric functions y(t) = -4t^2 + 16t and x(t) = -2t. Will Greg’s bowling ball reach the pins before the gutters?
- A particle is moving on the xy-plane. Its motion can be described by the parametric functions y(t) = sin(t) and x(t) = √t. Find the equation of the line tangent to particle at time t = 16π.
- You roll two marbles on a flat surface at time t = 0. The position of the first marble can be described by the parametric functions: y(t) = 3(√(t/2)) + 1 and x(t) =2t. The position of the second marble can be described by the parametric functions: y(t) = 2^t and x(t) = t^2. The marbles will meet at two distinct times. Find the equation of the line that passes through both points at which the marbles meet.
- A particle moves on the xy-plane. Its motion can be described by the parametric functions y(t) = sin(t) and x(t) = cos(t). What will the path of the particle look like?
- You ask four kindergarten students to scribble on a grid. The following pairs of parametric functions describe the path of their scribble as a function of time, starting at t=0. Which of the following will reach the point (1, 1) first.
- A rocket is launched from Earth. Its distance from Earth’s surface as a function of time is given by D(t) = 12t^2 + 40t. The amount of fuel in the tank as a function of distance is given by F(D) = 1000 - 0.1D. Convert distance and fuel into parametric functions with time, t, as a parameter and verify dF/dD.
- Greg is bowling with his friends and rolls the ball at time t = 0. Consider the center axis of the lane to correspond to line x = 0 and the pin deck to be at the line y = 15. The gutters correspond to the lines x = -4 and x = 4. If a ball falls into a gutter before hitting any pins, Greg’s score is 0. The motion of the ball can be described by the parametric functions y(t) = -4t^2 + 16t and x(t) = -2t. Find the parametric second derivative d^2y/dx^2 of the ball’s motion.
- A particle is moving on the xy-plane starting at t = 0. Its motion can be described by the parametric functions y(t) = 3t^3 - 2t and x(t) = ln(t). At what time, t, is the path of the particle neither concave up nor concave down?
- You roll two marbles on a flat surface at time t = 0. The position of the first marble can be described by the parametric functions: y(t) = 3(√(t/2)) + 1 and x(t) = 2t. The position of the second marble can be described by the parametric functions: y(t) = 2^t and x(t) = t^2. The marbles will meet at two distinct times. Evaluate the first derivative of the first marble at the first time and the second derivative of the second marble at the second time.
- A particle moves on the xy-plane. Its motion can be described by the parametric functions y(t) = sin(t) and x(t) = cos(t). For which t in the interval 0 ≤ t < 2π is the path of the particle concave down?
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