Problem 28 - College Trigonometry

Question

In Exercises \(25-39\), find a parametric description for the given oriented curve. \text { the curve } y=4-x^{2} \text { from }(-2,0) \text { to }(2,0) \text { . }

Step-by-Step Solution

Answer

Parametric equations: \(x(t) = t\), \(y(t) = 4 - t^2\), for \(t\in[-2, 2]\).

Step 1: Identify the Curve Equation

The problem provides the curve equation as \(y = 4 - x^2\). This is a parabola that opens downward.

Step 2: Determine the Range for x

The curve starts at the point \((-2, 0)\) and ends at \((2, 0)\). Therefore, the range for \(x\) is from \(-2\) to \(2\).

Step 3: Parametrize x in Terms of a Parameter t

Let \(t\) represent a parameter for the curve, where \(t\) ranges from \(-2\) to \(2\). We can set \(x(t) = t\), so that as \(t\) varies from \(-2\) to \(2\), \(x\) varies over the same interval.

Step 4: Parametrize y in Terms of t

Using the original curve equation \(y = 4 - x^2\), and substituting \(x = t\), we find \(y = 4 - t^2\). This gives the parameterization \(y(t) = 4 - t^2\).

Step 5: Write the Parametric Equations

We can express the curve using the parametric equations: \(x(t) = t\) and \(y(t) = 4 - t^2\), where \(t\) is in the interval \([-2, 2]\).

Key Concepts

Parametric Description

A parametric description changes the way we describe a curve in mathematics, shifting from traditional x and y forms to using a third variable or parameter, often denoted as \( t \). Instead of \( y \) being solely a function of \( x \), both \( x \) and \( y \) become functions of \( t \).
This approach can simplify the manipulation of curves across variable spaces and make it easier to describe movements or orientations along a path. In our exercise, \( x(t) = t \) and \( y(t) = 4 - t^2 \) act as the parametric equations. \( t \) smoothly varies from \(-2\) to \(2\), causing both \( x \) and \( y \) to trace the shape of the parabola efficiently.
The benefit of using parametric descriptions lies in its application to curves that aren't easily depicted in Cartesian coordinates or have a specific direction or flow along them. This is particularly useful for studying curves in motion or modeling objects in physics and engineering.

Parabola

The parabola is a specific type of curve represented by a quadratic equation, often written in the form \( y = ax^2 + bx + c \). In our case, the equation is \( y = 4 - x^2 \), which represents a downward-opening parabola controlled by the \(x^2\) term.

• The term \(-x^2\) indicates that the parabola opens downwards.
• The vertex of this parabola is at its maximum point, at \( (0, 4) \).
• The equation can describe sections of the curve, as exhibited in the problem, which focuses from \((-2, 0)\) to \((2, 0)\).

Parabolas have diverse applications, from physics, in contexts like projectile motion, to various branches of engineering and economics. The unique shapes and properties of parabolas allow them to be used for designing parabolic antennas and optimizing reflective surfaces.

Curve Orientation

Curve orientation is about the direction the curve travels as the parameter \( t \) progresses.
For example, in our parabola parametric equations, \( x(t) = t \) and \( y(t) = 4 - t^2 \), \( t \) moves from \(-2\) to \(2\).
This means that the curve starts at \((-2, 0)\), ascends to the vertex at \((0, 4)\), and then descends to \((2, 0)\).
The key aspects:

• Orientation can depict whether the curve travels left to right, right to left, or any other perceivable direction.
• Knowing the orientation is vital, especially in graphics and animations, to keep the movement of objects smooth and natural.
• In physics, understanding orientation helps in modeling dynamics where direction matters, like velocity.

Identifying orientations can illuminate the relationship between the parameterization and real-world scenarios, making visualization and practical application more coherent.

X-Y Plane

The x-y plane, or Cartesian plane, is a fundamental concept when working with curves and lines in two dimensions.
It consists of two perpendicular axes that intersect at a point called the origin, marked as \( (0, 0) \).

• The horizontal axis, also known as the x-axis, typically represents the independent variable or the input.
• The vertical axis, or y-axis, usually shows the dependent variable or the output.

Each point on the x-y plane corresponds to a unique pair of numbers, allowing efficient depiction of curves, lines, and geometric shapes.
By converting equations into parametric forms, we can visualize segments of curves like the parabola \( y = 4 - x^2 \) on this plane.
The conversion simplifies drawing paths or sections of curves between two points, evident in how our curve spans from \((-2, 0)\) to \((2, 0)\) on the x-y plane.
This foundational aspect of analytical geometry serves in diverse branches of science and mathematics for modeling scenarios, plotting data, and solving equations.