91Ó°ÊÓ

Linear equations represent straight lines in the coordinate plane. They take the general form of \(y = mx + c\) where \(m\) is the slope, and \(c\) is the y-intercept. In linear equations, the relationship between the variables is constant, meaning the graph of the equation is a straight line.

For the given parametric equations, we expressed \(y\) directly in terms of \(x\) and found \(y = x - 1\). This form is similar to \(y = mx + c\) with \(m = 1\) and \(c = -1\). Hence, the equation corresponds to a line.

• The slope \(m = 1\) indicates the line rises one unit for every unit it moves right.
• The y-intercept \(c = -1\) tells us the line crosses the y-axis at \(-1\).

This confirms that the parametric curve for these equations indeed forms a straight line.

A parametric curve is described by two or more equations where each component is expressed in terms of a common parameter, usually \(t\). The idea is to capture the change in the curve's position over a continuous interval of \(t\). In other words, parametric equations define how \(x\) and \(y\) coordinates position along a curve as \(t\) varies.

For the exercise at hand, the parametric equations:

• \(x = (3t + 2)^2\)
• \(y = (3t + 2)^2 - 1\)

are dependent on the parameter \(t\). By manipulating \(t\), the values of \(x\) and \(y\) change, moving the point on a coordinate plane accordingly.

Interestingly, in this case, when substituting \(y\) in terms of \(x\), we see it boils down to a simple linear relationship, defining a straight path or line. This indicates that while parametric equations can represent more complex curves, they can also define simple linear paths in some instances.

The range of the parameter \(t\) is vital in determining which part of the parametric curve is being considered. It dictates the start and end points of the curve. Here, our parameter \(t\) is defined to range from 0 to 3.

When \(t = 0\), substituting into \(x = (3t + 2)^2\) gives \(x = 4\). When \(t = 3\), \(x = (3t + 2)^2 = 121\). Hence, \(x\) ranges from 4 to 121 as \(t\) moves from 0 to 3. This continuous interval ensures the line is fully sketched between these positions without any gaps.

Defining the parameter range ensures that we can systematically trace out the segment of the curve we are interested in. In mathematical problems, especially involving parametric equations, knowing the range assists in correctly identifying or excluding parts of curves needed for analysis.