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

• In a parametric function, the parameter is the independent variable.
• In an equation, the independent variable is controlled, while the dependent variable changes based on its value.
• Order the parameter values and their corresponding points on the curve for x(t) = t + 1 and y(t) = t²

  1️⃣ t = 0, (1, 0)

  2️⃣ t = 1, (2, 1)

  3️⃣ t = 2, (3, 4)
• If x = t + 1 and y = t², as t increases from 0 to 2, the curve traced is a parabola
• In the equation T = 2t + 5, t is the independent variable.
• For x = t + 1, the derivative dx/dt is equal to 1
• A parametric function uses a third variable called the parameter
• Match the variable type with its role:

  Parameter ↔️ Independent variable

  Variables x and y ↔️ Dependent variables
• Changes in parameters dictate the shape and path of the parametric curve
• What is the independent variable in a parametric function called?

  Parameter
• What is the role of the independent variable in an equation?

  Controlled or chosen
• What are used to describe the rates of change in parametric functions?

  Derivatives
• In an equation, what is the term for a variable that depends on the value of another variable?

  Dependent variable
• Variables x and y in a parametric function are dependent on the parameter t.


  True
• Changes in parameters directly influence the quantities described by variables in a parametric function.

  True
• A parametric function expresses two variables, x and y, using a third variable called the parameter t.

  True
• Match the components of a parametric function with their descriptions:

  Parameter (t) ↔️ Independent variable

  Variables (x, y) ↔️ Dependent variables expressed in terms of t
• Changes in parameters influence the shape and path of the parametric curve.


  True
• For x = t + 1 and y = t², the rate of change of y with respect to t is 2t.


  True