Parametric Functions | undefined Unit 4 Study Guide

The Core Idea: Parametric Functions

In many contexts, describing a curve or the path of an object directly with a single equation relating $y$ to $x$ (like $y = f (x)$ ) is insufficient. This approach fails to capture information like the direction of motion or the position of an object at a specific moment in time. Parametric functions solve this by introducing a third variable, called a parameter, typically denoted by $t$ . Instead of defining $y$ in terms of $x$ , we define both $x$ and $y$ as separate functions of $t$ . This creates a set of equations, $x = f (t)$ and $y = g (t)$ , which together define a plane curve.

The parameter $t$ acts as an independent variable that generates the $(x, y)$ coordinates of points on the curve. As $t$ varies over a given interval, it traces out the path of the curve in the $x y$ -plane. It is crucial to understand that the parameter $t$ is not plotted as an axis on the graph; it is the underlying variable that controls the position $(x, y)$ . This framework is especially powerful for modeling the path of an object, where $t$ can naturally represent time, and the equations describe the object's horizontal and vertical positions at any given moment.

Key Definitions

The study of parametric functions is built on a few core definitions derived directly from how they are constructed.

Plane Curve: A plane curve is the set of all ordered pairs $(x, y)$ in a coordinate plane that are determined by the outputs of two functions, $x = f (t)$ and $y = g (t)$ , for a variable $t$ over a specified interval.
Parametric Equations: The two equations that define the plane curve, $x = f (t)$ and $y = g (t)$ , are known as the parametric equations for the curve.
Parameter: The parameter, denoted by $t$ , is the independent variable in the parametric equations. The value of the parameter determines the specific coordinates $(x, y)$ of a point on the curve. The parameter itself is not represented by an axis in the $x y$ -plane.

Understanding The Role of the Parameter

The most significant conceptual shift when working with parametric functions is understanding the role of the parameter, $t$ . In a standard function $y = f (x)$ , there is a direct relationship between the horizontal and vertical coordinates. In a parametric representation, this direct link is replaced by a common dependency on $t$ .

The parameter $t$ dictates the position of a point $(x, y)$ on the curve. For each specific value of $t$ in a given interval, the equations $x = f (t)$ and $y = g (t)$ produce exactly one $x$ -coordinate and one $y$ -coordinate, defining a single point on the plane. As the value of $t$ increases, a sequence of points is generated, tracing the curve in a specific direction. This inherent directionality, or orientation, is a key feature of parametric curves.

Think of this in the context of modeling an object's path, as mentioned in the essential knowledge. If $t$ represents time in seconds, $t = 0$ gives the starting position $(x (0), y (0))$ , $t = 1$ gives the position after one second $(x (1), y (1))$ , and so on. Connecting these points in the order they are generated reveals not just the shape of the path, but the direction in which the object traveled along that path. The parameter provides a dynamic description of the curve being traced, rather than a static picture of its shape.

Core Concepts & Rules

A curve in the $x y$ -plane is defined by a pair of parametric equations, $x = f (t)$ and $y = g (t)$ , where both $x$ and $y$ are functions of a common parameter, $t$ .
The parameter $t$ is an independent variable that generates the $(x, y)$ coordinates. It is not plotted as a physical axis on the $x y$ -plane graph of the curve.
For any given value of $t$ within a defined interval, the parametric equations produce a single point $(x, y)$ that lies on the curve.
Parametric equations are a natural way to model the path of a moving object, with the parameter $t$ often representing time. The equations then specify the object's position at any given time.

Step-by-Step Example 1: Plotting a Basic Parametric Curve

Problem: Sketch the plane curve defined by the parametric equations $x (t) = 2 t - 1$ and $y (t) = t^{2} + 1$ for the parameter interval $0 \leq t \leq 3$ .

Step 1: Create a table of values for $t$ , $x$ , and $y$ .

Choose several values for $t$ within the given interval $[0, 3]$ . It is often helpful to start with the endpoints and include some integer values in between.

$t$	$x (t) = 2 t - 1$	$y (t) = t^{2} + 1$	$(x, y)$
0	$2 (0) - 1 = - 1$	$(0)^{2} + 1 = 1$	$(- 1, 1)$
1	$2 (1) - 1 = 1$	$(1)^{2} + 1 = 2$	$(1, 2)$
2	$2 (2) - 1 = 3$	$(2)^{2} + 1 = 5$	$(3, 5)$
3	$2 (3) - 1 = 5$	$(3)^{2} + 1 = 10$	$(5, 10)$

Step 2: Plot the $(x, y)$ coordinate pairs.

Plot the points $(- 1, 1)$ , $(1, 2)$ , $(3, 5)$ , and $(5, 10)$ on the $x y$ -plane.

Step 3: Connect the points to form the curve.

Draw a smooth curve that passes through the plotted points, starting from the point corresponding to $t = 0$ and ending at the point corresponding to $t = 3$ .

Step 4: Indicate the orientation (direction) of the curve.

Draw small arrows along the curve to show the direction it is traced as $t$ increases from 0 to 3. The curve starts at $(- 1, 1)$ and moves towards $(5, 10)$ .

The final sketch will show a segment of a parabola opening upwards, starting at $(- 1, 1)$ and ending at $(5, 10)$ , with arrows indicating this direction of travel.

Step-by-Step Example 2: Interpreting a Parametric Model

Problem: The path of a drone is modeled by the parametric equations $x (t) = 8 t$ and $y (t) = - 16 t^{2} + 64 t$ , where $t \geq 0$ is the time in seconds, and $x$ and $y$ are the horizontal and vertical distances in feet, respectively. Find the location of the drone at $t = 2$ seconds.

Step 1: Identify the given parameter value.

The problem asks for the drone's location at a specific time, which is the parameter. We are given $t = 2$ .

Step 2: Substitute the parameter value into the equation for $x (t)$ .

This will give the horizontal coordinate of the drone at $t = 2$ .

$x (2) = 8 (2) = 16$

Step 3: Substitute the parameter value into the equation for $y (t)$ .

This will give the vertical coordinate of the drone at $t = 2$ .

$y (2) = - 16 (2)^{2} + 64 (2)$

$y (2) = - 16 (4) + 128$

$y (2) = - 64 + 128 = 64$

Step 4: State the final coordinates.

The coordinates of the drone at $t = 2$ seconds are $(x (2), y (2))$ .

Based on the calculations, the drone is at the point $(16, 64)$ . This means it is 16 feet horizontally from its starting point and 64 feet vertically from its starting point.

Using Your Calculator

A graphing calculator is an essential tool for visualizing the plane curve defined by a set of parametric equations.

To graph a parametric curve (e.g., $x (t) = 2 t - 1$ , $y (t) = t^{2} + 1$ for $0 \leq t \leq 3$ ):

Set the Mode:
- Press the [MODE] button.
- Navigate down to the line that reads FUNCTION (or FUNC).
- Use the arrow keys to highlight PARAMETRIC (or PAR) and press [ENTER].
Enter the Equations:
- Press the [Y=] button. You will now see input fields for pairs of equations: $X_{1 t}$ , $Y_{1 t}$ , $X_{2 t}$ , etc.
- In $X_{1 t}$ , enter the expression for $x (t)$ . For this example: 2T - 1. (The $[T]$ variable is typically accessed using the same button as $[X]$ , often labeled $[X, T, θ, n]$ ).
- In $Y_{1 t}$ , enter the expression for $y (t)$ . For this example: $T^{2} + 1$ .
Set the Window:
- Press the [WINDOW] button. This is the most critical step for parametrics.
- $T min$ : Set the starting value of the parameter. For this example, $T min = 0$ .
- $T ma x$ : Set the ending value of the parameter. For this example, $T ma x = 3$ .
- $T s t e p$ : This determines how often the calculator plots a point. A smaller value gives a smoother curve. A good starting point is $0.1$ or $(T ma x - T min) /100$ . Let's use $T s t e p = 0.1$ .
- $X min$ , $X ma x$ , $Ymin$ , $Yma x$ : Set the viewing window for the $x y$ -plane. Based on our example table, we need $x$ to go from at least -1 to 5 and $y$ from 1 to 10. A good window would be $X min = - 2$ , $X ma x = 6$ , $Ymin = 0$ , $Yma x = 12‘.4. * * G r a p han d T r a ce : * * * P ress ‘ [GR A P H] ‘. T h ec a l c u l a t or w i ll d r a wt h ec u r v e . * P ress ‘ [TR A CE]$ . You can now use the left and right arrow keys to move a cursor along the curve. As you move, the calculator will display the corresponding values of $t$ , $x$ , and $y$ at the bottom of the screen. This is a powerful way to see exactly how the parameter $t$ generates the $(x, y)$ points on the curve.

AP Exam Quick Hit

Common Question Types

Finding a Point at a Specific Parameter Value: Given parametric equations $x (t)$ and $y (t)$ and a specific value $t = c$ , you will be asked to find the coordinates $(x, y)$ of the point on the curve.
- Example: "A particle's position is given by $x (t) = t^{2} - 3$ and $y (t) = 4 t + 1$ . What is the position of the particle at $t = 2$ ?"
Identifying a Graph: You will be presented with a set of parametric equations and several graphs, and you must choose the graph that correctly represents the curve, including its starting point and direction.
- Example: "Which of the following graphs shows the curve defined by $x (t) = 3 cos (t)$ and $y (t) = 2 sin (t)$ for $0 \leq t \leq π$ ?"
Interpreting a Model: Given a real-world scenario modeled by parametric equations (e.g., the path of a projectile), you will be asked to interpret the meaning of the coordinates at a certain time or find the time when an object reaches a specific location.
- Example: "The path of a boat is modeled by $x (t) = 10 t$ and $y (t) = 4$ . At what time $t$ is the boat at the point $(50, 4)$ ?"

Common Mistakes

Plotting the Parameter: A frequent error is attempting to plot $t$ as one of the axes, for instance, by graphing $(t, x)$ or $(t, y)$ . Remember, the graph is only in the $x y$ -plane; $t$ is an independent variable that generates the points.
Ignoring the Parameter Interval: Failing to restrict the graph to the specified interval for $t$ . If a curve is defined for $0 \leq t \leq 4$ , your graph should only show the portion of the curve traced out during that interval, not the entire potential curve.
Omitting Direction/Orientation: Simply drawing the shape of the curve without indicating its direction. Parametric curves have an inherent orientation as $t$ increases, which must be shown with arrows on the curve.
Incorrect Calculator Window Settings: Using inappropriate $T min$ , $T ma x$ , or $T s t e p$ values in the calculator. A $T s t e p$ that is too large can make a smooth curve appear as a series of disconnected straight lines. Incorrect $X$ and $Y$ window settings may cause the curve to be partially or completely off-screen.

Parametric Functions - AP PreCalculus Study Guide