Problem 24 - 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. $$x=\frac{6}{5} \sqrt{25-t^{2}}, \quad y=t, \quad|t| \leq 5$$

Step-by-Step Solution

Answer

Equation: \( 25x^2 + 36y^2 = 900 \). Identify as an ellipse with orientation from \( y = -5 \) to \( y = 5 \).

Step 1: Recognize Parametric Equations

The given equations are parametric equations of a curve, with parameter \( t \). The equations are \( x = \frac{6}{5} \sqrt{25 - t^2} \) and \( y = t \), with \(|t| \leq 5\). Our goal is to eliminate the parameter \( t \) to find an equation solely in terms of \( x \) and \( y \).

Step 2: Solve for t in Terms of y

From the second equation, \( y = t \), we directly get \( t = y \). This expression will allow us to eliminate \( t \) from the equations.

Step 3: Substitute t with y in the First Equation

Substitute \( t = y \) into the first parametric equation: \( x = \frac{6}{5} \sqrt{25 - y^2} \). This gives us a relationship between \( x \) and \( y \).

Step 4: Square x to Isolate y

Square both sides of the equation \( x = \frac{6}{5} \sqrt{25 - y^2} \) to eliminate the square root:\[ x^2 = \left( \frac{6}{5} \right)^2 (25 - y^2) \]Simplify to get:\[ 25x^2 = 36(25 - y^2) \]

Step 5: Simplify to Find the Cartesian Equation

Expand and simplify the equation:\[ 25x^2 = 900 - 36y^2 \]Rearrange this to the form:\[ 25x^2 + 36y^2 = 900 \]This is the equation of an ellipse.

Step 6: Identify Limits and Orientation

Notice that \(|t| \leq 5\) imposes \( -5 \leq y \leq 5 \), as \( y = t \). The curve is oriented from \( y = -5 \) to \( y = 5\) as \( t \) increases from \(-5 \) to \( 5\).

Step 7: Sketch the Graph

Using the derived Cartesian equation \( 25x^2 + 36y^2 = 900 \), recognize it as an ellipse centered at the origin with semi-axis lengths given by \( a = 6 \) (for \( x \)) and \( b = 5 \) (for \( y \)). The orientation can be marked on the graph from bottom to top along the \( y \)-axis.

Key Concepts

Ellipse

An ellipse is a closed curve on a plane that surrounds two focal points. Imagine it as a "squished circle". It's quite different from a circle because the distance is not constant from the center to the edge.
Instead, the sum of the distances from any point on the curve to the two foci (fixed points) is constant.
This property of ellipses makes them arise naturally in many physical systems and geometrical configurations.

• In our specific problem, the equation that results is that of an ellipse: \( 25x^2 + 36y^2 = 900 \).
• Recognizing an ellipse involves noting the presence of squared terms with different coefficients in the Cartesian equation.

Understanding what makes an ellipse unique is the first step toward analyzing the curve correctly.

Cartesian Equation

The process of converting parametric equations into a Cartesian equation is one of the keystones of understanding how these mathematical expressions relate to their graph.

• Parametric equations are usually in terms of a parameter, in this case, \( t \). They are expressed in terms of two separate equations for \( x \) and \( y \).
• To convert them into a Cartesian equation, we aim to eliminate the parameter \( t \).

In this example, we are given: \( x = \frac{6}{5} \sqrt{25 - t^2} \) and \( y = t \). By substituting \( t \) in terms of \( y \), we derived \[ 25x^2 + 36y^2 = 900 \].
This means we now have a single equation that relates \( x \) and \( y \), without the parameter \( t \).
The transition to a Cartesian equation often involves algebraic manipulation such as squaring to eliminate square roots, just as we did here.

Graph Sketching

When sketching the graph of an ellipse, it’s important to know several key characteristics. An ellipse is symmetric across both its major and minor axes.

• This equation \( 25x^2 + 36y^2 = 900 \) suggests it is centered at the origin \((0,0)\).
• Identifying the semi-major and semi-minor axes is crucial: Here, \( a = 6 \) and \( b = 5 \).
• These values correspond to the extent of the ellipse along the \( x \)-axis and \( y \)-axis, respectively.

To accurately sketch, mark the extremities at \( (6,0) \), \( (-6,0) \), \( (0,5) \), and \( (0,-5) \) on a Cartesian plane. Since the ellipse is symmetric, ensure it is even and rounded through these points.
A neatly drawn ellipse helps visualize the relationships between the components of the parametric equations.

Orientation

Orientation of a curve describes the direction in which the curve moves as the parameter \( t \) increases. For ellipses described by parametric equations, this can indicate how the ellipse is "traced" over time.
In our problem, \( y = t \), so as \( t \) moves from \(-5 \) to \( 5 \), \( y \) does the same.

• This means the ellipse is traced from bottom (\( y = -5 \)) to top (\( y = 5 \)).
• The orientation can be represented on the sketch by marking an arrow showing the direction of increase of \( y \).

By paying attention to orientation, you understand more than just the shape and position of the graph — you also capture the dynamic element of how it's drawn.