Problem 23 - Precalculus: Functions and Graphs

Question

Find an equation in \(x\) and \(y\) whose graph contains the points on the curve \(C\). Sketch the graph of \(C\), and indicate the orientation. $$x=-2 \sqrt{1-t^{2}}, y=t, \quad|t| \leq 1$$

Step-by-Step Solution

Answer

The equation is \(x^2 + 4y^2 = 4\); it's an ellipse. Orientation: bottom to top.

Step 1: Understand the Parametric Equations

The given parametric equations are \(x = -2 \sqrt{1-t^2}\) and \(y = t\). These equations define a curve using the parameter \(t\), where \(|t| \leq 1\). Our task is to find a single equation relating \(x\) and \(y\) only.

Step 2: Express \(t^2\) in terms of \(y\)

Since \(y = t\), we have \(t = y\). Therefore, \(t^2 = y^2\). This simplifies our expression for \(x\).

Step 3: Substitute \(t^2 = y^2\) into \(x\) Equation

Substitute the expression \(t^2 = y^2\) into the equation for \(x\): \[x = -2 \sqrt{1 - t^2} = -2 \sqrt{1 - y^2}\].

Step 4: Square Both Sides to Eliminate the Square Root

To eliminate the square root, square both sides of the equation: \[x^2 = (-2 \sqrt{1 - y^2})^2 = 4(1 - y^2)\].

Step 5: Simplify the Equation

Simplify the equation obtained in Step 4: \[x^2 = 4 - 4y^2\]. We can rewrite this as \[x^2 + 4y^2 = 4\].

Step 6: Identify the Graph and Orientation

The equation \(x^2 + 4y^2 = 4\) represents an ellipse centered at the origin. The parameter \(|t| \leq 1\) indicates that the curve traces from \(y = -1\) to \(y = 1\), since as \(t\) goes from -1 to 1, the curve is traced once from the bottom to the top.

Key Concepts

Ellipses

In mathematics, an ellipse is a set of points such that the sum of the distances from two fixed points (called foci) is constant. Ellipses appear in various mathematical contexts, such as geometry and calculus.
In our case, we are dealing with an ellipse represented by the equation \(x^2 + 4y^2 = 4\). This equation is a standard form of an ellipse centered at the origin. To understand the shape, consider that the equation \(\frac{x^2}{4} + \frac{y^2}{1} = 1\) is derived from the given ellipse equation. Here, the lengths of the semi-major and semi-minor axes can be identified.

• The semi-major axis corresponds to the larger denominator: 2, since \(\sqrt{4} = 2\).
• The semi-minor axis corresponds to the smaller denominator: 1, since \(\sqrt{1} = 1\).

With these axes, the ellipse is stretched along the \(x\)-axis horizontally, having a longer stretch of 2 units, and compressed along the \(y\)-axis vertically, having a shorter stretch of 1 unit. This visual representation and understanding of the axes help us comprehend the orientation of the ellipse curve.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe the Cartesian coordinate system. It connects geometric properties and relationships with algebraic equations, providing insights into shapes, distances, and angles.
In the given exercise, converting the parametric equations \(x = -2 \sqrt{1-t^2}\) and \(y = t\) into a coordinate form \(x^2 + 4y^2 = 4\) helps reveal the geometric nature of the curve as an ellipse. This showcases the power of coordinate geometry in describing and analyzing geometric figures using algebraic methods.

• By expressing one variable in terms of another, we transform parametric forms into algebraic forms.
• We can analyze the curve's properties such as its shape, position, and orientation using these transformed equations.

Understanding this link between equations and shapes sharpens one's ability to visualize and solve geometrical problems via algebraic techniques.

Curve Tracing

Curve tracing is the process of following the path of a curve in the coordinate plane to understand its direction, shape, and behavior. This technique is crucial in visualizing how a curve behaves over an interval based on the parametric or standard equations.
With respect to our ellipse given by the equation \(x^2 + 4y^2 = 4\), tracing the curve involves sketching the ellipse based on its parametric equations \(x = -2 \sqrt{1-t^2}\) and \(y = t\), where \(|t| \leq 1\). As \(t\) progresses from -1 to 1, it effectively traces out the path of the ellipse from the bottom to the top.

• To start, note the ellipse boundaries: \(y = -1\) to \(y = 1\).
• The curve's direction is determined by the increasing values of \(t\) from negative to positive.

This tracing reflects the path of motion on the ellipse, which is essential for understanding not just static shapes but dynamic processes in coordinate geometry.