Learning objective 4.1.A
Construct a graph or table of values for a parametric function represented analytically.
The coordinates (x_{i},\, y_{i}) at time t_{i} can be written as functions of t and can be expressed as the single parametric function f(t) = (x(t),\, y(t)), where in this case x and y are names of two functions. For example, consider the parametric function f(t) = (x(t),\, y(t)) where x(t) = t^2 and y(t) = 2t. In this parametric function, x and y are dependent on the parameter t. As t changes, the values of x and y change according to their respective functions: x(t) = t^{2} \\ y(t) = 2t
The parametric function: f(t)=(x(t),\, y(t)). Replace x(t) and y(t) in the expression f(t) = (x(t),\, y(t)), with the provided functions. The parametric function f(t) for the given x(t) and y(t) functions is f(t) = (t^{2},\, 2t).
Given x(t) = 3t - 1 and y(t) = 2t + 1, express the parametric function f(t).
Parametric functions involve two dependent variables, x and y, dependent on a single independent variable, t.
A graph of a parametric function can be created by plotting the points from the table of values and connecting them in order of increasing values of t. Visualizing parametric functions through graphs allows us to better understand the behavior and properties of these functions, making it easier to analyze and interpret the underlying relationships.
Given the parametric function f(t) = (x(t),\, y(t)) where x(t) = t^{2} and y(t) = 2t, create a table of values and sketch the graph of the function for t in the interval [0,\, 3].
To create a table of values for a parametric function f(t) = (x(t),\, y(t)):
Choose several values of t within the domain of the function.
Calculate the corresponding x and y values using the given functions.
Fill in the table with the calculated values.
The parametric function: f(t)=(x(t),\, y(t))
f(t) = (t^{2},\, 2t)
t | x | y |
---|---|---|
0 | 0 | 0 |
1 | 1 | 2 |
2 | 4 | 4 |
3 | 9 | 6 |
Sketch the graph of a parametric function f(t)=(x(t),\, y(t)) in the interval [0,\, 3]:
Plot the points from the table on a coordinate plane.
Connect the points in order of increasing value of t.
Given the parametric function f(t) = (3t - 1,\, 2t + 1) with t in the domain [0,\, 3].
Create a table of values.
t | x | y |
---|---|---|
0 | ||
1 | ||
2 | ||
3 |
Graph the parametric function.
A table of values can be created along with a sketch of the graph for parametric functions to be represented analytically. This will help them analyze and understand the behavior of the functions in different contexts.