Parametric Function
Definition
A parametric function describes a set of equations that define coordinates as functions of an independent parameter (usually time). Instead of expressing y directly in terms of x, both x and y are expressed separately as functions with respect to another variable.
Analogy
Think about playing catch with a friend using two different balls thrown at different speeds and angles. Each ball follows its own path independently but can still be described by separate equations based on time. One equation determines how far horizontally one ball travels over time (x-coordinate), while another equation determines its vertical position (y-coordinate).
Related terms
Parameterization: Parameterization refers to finding equations that describe variables in terms of an independent parameter such as time. It allows us to represent complex curves or motions using simpler equations.
Eliminating the Parameter: Sometimes, we want to express a parametric function in terms of x and y directly, without the parameter. This process is called eliminating the parameter and involves solving for x and y in terms of the parameter.
Symmetry: Parametric functions can exhibit various types of symmetry, such as horizontal symmetry (reflection across the x-axis) or vertical symmetry (reflection across the y-axis). Analyzing symmetries helps understand the behavior of parametric curves.
"Parametric Function" appears in:
Study guides (2)
Practice Questions (7)
What is the purpose of finding the second derivative of a parametric function?
Which of the following formulas yields the second derivative, d^2y/dx^2, of a parametric function?
You have developed a new strain of bacteria in your lab. You have made DNA modifications that allow it to become resistant to an antibiotic. However, a side effect of the antibiotic is the inability to reproduce as quickly as the original strain. You find that the number of births in the colony as a function of time is B(t) = √t. Additionally, you find that the number of deaths in the colony as a function of time can be modeled by the parametric function D(t) = ln(t). Find the value of the parametric second derivative, d^2B/dD^2, at time t = 2.
What is a parametric function in the context of AP Calculus?
What is the displacement of a parametric function defined by x(t) and y(t)?
What does the integral from a to b of v(t) represent in the context of a parametric function?
What does the displacement of a parametric function tell us?
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