91Ó°ÊÓ

Parametric equations offer a unique way to describe curves in the coordinate plane. Unlike standard Cartesian equations, which relate \( x \) and \( y \) directly, parametric equations use a parameter, usually denoted as \( t \), to express both \( x \) and \( y \). For example, in our exercise, the equations are \( x = 3t \) and \( y = t^2 - 1 \). These equations allow each point on the curve to be represented as a function of \( t \). This method effectively describes curves that may be difficult to express with a single equation in \( x \) and \( y \). Parametric equations can make it easier to sketch complex shapes and patterns, like spirals or ellipses, by manipulating the parameter \( t \).

Graphing parametric curves involves calculating specific coordinates for different values of the parameter \( t \). In our given problem, the range for \( t \) is from 0 to 4. We need to compute both \( x \) and \( y \) coordinates for specific \( t \) values to plot and sketch the curve correctly. To start, compute the points as shown:

• For \( t=0 \), \( (x, y) = (0, -1) \)
• For \( t=1 \), \( (x, y) = (3, 0) \)
• For \( t=2 \), \( (x, y) = (6, 3) \)
• For \( t=3 \), \( (x, y) = (9, 8) \)
• For \( t=4 \), \( (x, y) = (12, 15) \)

By plotting these points on a coordinate graph and connecting them with a smooth curve, we reveal the path traced as \( t \) changes. The shape of this particular curve is a parabola, and understanding the pattern helps discern its geometric properties.

The direction of a parametric graph indicates how the curve progresses as the parameter \( t \) changes. For any parametric equation, observing how \( x \) and \( y \) change as \( t \) increases is key to identifying the curve's direction. In this exercise, as \( t \) varies from 0 to 4:

• \( x \) moves from 0 to 12
• \( y \) moves from -1 to 15

This tells us the graph moves from left to right. Representing the direction on a plot is crucial for properly understanding the motion along the path. To effectively show this, draw arrows on the curve pointing in the increasing \( t \) direction from the first computed point \((0, -1)\) to the last \((12, 15)\). These arrows offer a visual guide indicating how the graph unfolds with consecutive \( t \) values.

Key points on a parametric curve serve as guideposts for accurately drawing the curve. To find key points, substitute several distinct values of \( t \) within the defined range (0 to 4 in our situation) into the parametric equations \( x = 3t \) and \( y = t^2 - 1 \). Record the \( (x, y) \) pairs calculated:

• \( t = 0 \) leads to the point \((0, -1)\)
• \( t = 1 \) leads to the point \((3, 0)\)
• \( t = 2 \) leads to the point \((6, 3)\)
• \( t = 3 \) leads to the point \((9, 8)\)
• \( t = 4 \) leads to the point \((12, 15)\)

These coordinates outline the progression of the curve and are then plotted and connected in sequence. Understanding these key points is fundamental for constructing an accurate representation of the graph and identifying any special features, like turning points or intersections that might occur.