15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.


x = √t + 4, y = 3√t; 0 ≤ t ≤ 16

Verified step by step guidance

1. Identify the given parametric equations: \(x = \sqrt{t} + 4\) and \(y = 3\sqrt{t}\) with the parameter range \(0 \leq t \leq 16\).
2. Express \(\sqrt{t}\) from one of the equations to eliminate the parameter. For example, from \(y = 3\sqrt{t}\), solve for \(\sqrt{t}\): \(\sqrt{t} = \frac{y}{3}\).
3. Substitute \(\sqrt{t} = \frac{y}{3}\) into the equation for \(x\): \(x = \frac{y}{3} + 4\).
4. Rearrange the equation to express \(y\) in terms of \(x\): multiply both sides by 3 and isolate \(y\) to get \(y = 3(x - 4)\), which is the Cartesian equation of the curve.
5. To describe the curve, recognize that it is a straight line with slope 3 and y-intercept at \(y = -12\). The parameter \(t\) increases from 0 to 16, so \(\sqrt{t}\) increases from 0 to 4, meaning \(x\) increases from 4 to 8 and \(y\) increases from 0 to 12. This indicates the positive orientation is from the point \((4,0)\) to \((8,12)\) along the line.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves and motions.

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove t, resulting in a direct relationship between x and y. This process helps to identify the Cartesian equation of the curve, making it easier to analyze its shape.

Curve Orientation and Domain

The orientation of a parametric curve is determined by the direction in which the parameter t increases. Understanding the domain of t is essential to describe the portion of the curve traced and to indicate the positive direction along the curve.