Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine how the plane curves differ from each other. (a) \(x=2 \sqrt{t}\) \(y=4-\sqrt{t}\) (b) \(x=2 \sqrt[3]{t}\) \(y=4-\sqrt[3]{t}\) (c) \(x=2(t+1)\) \(y=3-t\) (d) \(x=-2 t^{2}\) \(y=4+t^{2}\)

Step-by-Step Solution

Answer

(a) Represents a portion of a parabola opening rightwards, traversed from top to bottom. (b) Shows a part of a cubic curve traversed entirely. (c) Gives a straight line traversed from right to left. (d) Displays a parabola opening to the left, traversed from bottom to top.

Step 1: Identify the Curves for Each Set of Parameters

The first step is to identify the type of curve each pair of parametric equations represent. We can do this by either plotting the curves or rewriting the parametric equations in Cartesian form if possible.

Step 2: Analyzing the Curves for (a) \(x=2 \sqrt{t}\) , \(y=4-\sqrt{t}\)

Plotting these parametric equations, the curve is a portion of a parabola opening to the right. The equations also can be written in the form \(y=4-\frac{x}{2}\). As \(t\) increases from 0, \(x\) increases while \(y\) decreases, so the parabola is traversed from top to bottom.

Step 3: Analyzing the Curves for (b) \(x=2 \sqrt[3]{t}\) , \(y=4-\sqrt[3]{t}\)

Plotting this set gives a curve which is a portion of a cubic function. The Cartesian form here is \(y=4-\frac{x^{3}}{8}\). The curve is traversed as \(t\) increases from \(-\infty\) to \(+\infty\).

Step 4: Analyzing the Curves for (c) \(x=2(t+1)\) , \(y=3-t\)

Here, the set of parametric equations represents a line when plotted. The Cartesian form of the equation is \(y=5-x\). As \(t\) ranges from \(-\infty\) to \(+\infty\), the line is traversed from right to left.

Step 5: Analyzing the Curves for (d) \(x=-2 t^{2}\) , \(y=4+t^{2}\)

This set of equations results in a parabola opening to the left when plotted. In the form \(y = 4 + 0.25x^{2}\), as \(t\) runs from \(-\infty\) to \(+\infty\), the parabola is traversed from bottom to top.

Key Concepts

Plane Curves

Plane curves are curves that exist on a two-dimensional plane. They can be described using parametric equations, where both the x and y coordinates are expressed as functions of a third variable, usually denoted as parameter "t". Parametric equations offer the flexibility to describe a wide variety of curves, some of which can be complex or impossible to depict with a single-function Cartesian equation.

• Each pair of parametric equations describes a path or locus of points in the plane.
• The parameter "t" often represents time or another progressive variable, allowing the natural or physical movement along the curve to be represented smoothly.
• By changing "t's" range, different segments or directions of the curves can be described.

Plane curves, including parabolas, ellipses, and hyperbolas, are foundational concepts in coordinate geometry, making them essential for understanding motion and other phenomena in physics and engineering.

Parabolas

A parabola is a specific type of plane curve that can be represented in parametric and Cartesian forms. In the context of plane geometry, a parabola is defined as the set of all points that are equidistant from a point called the "focus" and a line called the "directrix".
When using parametric equations, a parabola can take various forms depending on its orientation. For example:

• The parametric form - \(x = 2\sqrt{t}\) - \(y = 4 - \sqrt{t}\) represents a parabola that opens to the right.
• The Cartesian conversion of parametric equations often simplifies the analysis: for example, \(y = 4 - \frac{x}{2}\) is a horizontal parabola.

Parabolas are commonly seen in projectile motion, optics, and quadratic functions, which makes understanding their properties crucial in many fields of study.

Cartesian Form

The Cartesian form, also known as the Cartesian equation, explains curves explicitly with the "y" as a function of "x". It is different from the parametric form, where both coordinates are given as functions of one or more parameters.

• Converting parametric equations to Cartesian form provides a more traditional view of the curve.
• For example, the parametric equations \(x = 2\sqrt[3]{t}\) and \(y = 4 - \sqrt[3]{t}\) convert to the Cartesian form \(y = 4 - \frac{x^3}{8}\).
• This form helps in understanding the curve's behavior at a glance—as a single function without extra parameters.

Using Cartesian forms allows for easier differentiation and integration, aiding in graphing the curve's shape and studying its properties more comprehensively.

Cubic Functions

Cubic functions are functions of the form \(y = ax^3 + bx^2 + cx + d\), where "a," "b," "c," and "d" are constants, and "a" is non-zero. They are polynomial equations of degree three, which means they can produce S-shaped curves with potentially three real roots.

• A cubic function can also be expressed parametrically, which can help detail its path more dynamically.
• From the solution example, \(x = 2 \sqrt[3]{t}\) and \(y = 4 - \sqrt[3]{t}\) describe a cubic curve converted to the Cartesian form of \(y = 4 - \frac{x^3}{8}\).
• This interpretation reveals the nature of cubic behavior as "t" varies from negative to positive infinity.

Cubic functions appear in numerous analyses such as volumetric changes, paths of motion, and as part of Taylor and Fourier series, making them very versatile in mathematics.