4.1 Describing how quantities change with respect to each other in a parametric function

• A parametric function defines a curve or surface using a parameter, typically denoted as t
• In a parametric function, the x-coordinate is expressed as x(t).
• What curve is created by the parametric function x(t) = 2t and y(t) = t^2?

  Parabola
• The parameter in a parametric function dictates the position on the curve
• Parametric functions offer greater flexibility than standard functions like y = f(x).
• Steps to identify key components of a parametric function

  1️⃣ Identify the parameter (t)

  2️⃣ Define x(t) function

  3️⃣ Define y(t) function
• Match the parametric component with its description:

  Parameter (t) ↔️ Dictates the position on the curve

  x(t) ↔️ Defines the x-coordinate

  y(t) ↔️ Defines the y-coordinate
• Varying the parameter t traces the path of the curve in a parametric function.
• A parametric function uses a parameter typically denoted as t
• In parametric functions, the parameter t is often linear or quadratic.
• Changes in the parameter t in a parametric function affect the coordinates x(t) and y(t).
• Match the effect on coordinates with the corresponding change in parameter t:

  x(t) increases linearly ↔️ Moves rightward

  y(t) increases quadratically ↔️ Moves upward
• The rate of change in x(t) for the parametric function x(t) = 2t is constant.
• The parameter in a parametric function is the variable that dictates the position on the curve
• In a parametric function, the parameter t dictates the position along the curve or surface.
• What is a parametric function used to define?

  Curve or surface
• A parametric function expresses coordinates (x, y) as functions of a variable called a parameter
• Parametric functions offer greater flexibility in describing complex curves compared to standard functions.
• What does the parameter 't' dictate in a parametric function?

  Position on the curve
• In a parametric function, the x-coordinate is defined by the function x(t)
• The parameter 't' determines the position along the curve or surface in a parametric function.
• What is the role of y(t) in a parametric function?

  Defines the y-coordinate
• By varying the parameter 't', you can trace the path of the curve
• Match the parametric function components with their effects:

  t ↔️ Parameter

  x(t) ↔️ x-coordinate

  y(t) ↔️ y-coordinate
• What happens to the x-coordinate when 't' increases in the parametric function x(t) = 2t?

  Increases linearly
• In the parametric function x(t) = 2t and y(t) = t^2, the direction of movement as 't' increases is rightward for x and upward for y
• In the parametric function x(t) = 2t, the x-coordinate increases at a constant rate as 't' increases.
• What type of curve is traced by the parametric function x(t) = 2t and y(t) = t^2?

  Parabola
• Steps to visualize a parametric function on a graph:

  1️⃣ Plot the x(t) and y(t) functions separately

  2️⃣ Trace the path of the curve

  3️⃣ Observe the relationship between x and y
• Plotting x(t) = 2t results in a straight line with a constant slope
• When x(t) = 2t and y(t) = t^2 are traced, they create a parabolic curve in the xy-plane.