Problem 34 - Precalculus: Functions and Graphs

Question

Curves \(C_{1}, C_{2}, C_{3},\) and \(C_{4}\) are given parametrically, for \(t\) in \(\mathbb{R}\). Sketch their graphs, and indicate orientations. $$\begin{aligned}&C_{1}: x=t, \quad y=1-t\\\&C_{2}: x=1-t^{2}, \quad y=t^{2}\\\&C_{3}: x=\cos ^{2} t, \quad y=\sin ^{2} t\\\&C_{4}: x=\ln t-t, \quad y=1+t-\ln t, t>0\end{aligned}$$

Step-by-Step Solution

Answer

The graphs of \(C_1\), \(C_2\), \(C_3\), and \(C_4\) have orientations: \(C_1\) is linear, \(C_2\) is a leftward arc, \(C_3\) is counterclockwise in the first quadrant, and \(C_4\) is right to left.

Step 1: Analyzing Curve C1

Curve \(C_1\) is given by \(x = t\) and \(y = 1-t\). This is a straight line in the xy-plane. For different values of \(t\), \(x = t\) and \(y = 1 - t\) produce points like (0,1), (1,0), (2,-1), etc. The line moves to the right (as \(x\) increases) and downward (as \(y\) decreases) as \(t\) increases. The orientation is from top-left to bottom-right.

Step 2: Analyzing Curve C2

Curve \(C_2\) is given by \(x = 1-t^2\) and \(y = t^2\). This can be transformed into a parabola because as \(t\) varies, \(x\) changes due to \(1-t^2\) and \(y\) changes due to \(t^2\). Typical points are (1,0) for \(t = 0\), (0.5, 0.25) for \(t = 0.5\), and (0,1) for \(t = ±1\). The curve moves from right to left as \(t\) increases from -∞ to ∞.

Step 3: Analyzing Curve C3

Curve \(C_3\) is given by \(x = \cos^2(t)\) and \(y = \sin^2(t)\). Using the trigonometric identity \(\cos^2(t) + \sin^2(t) = 1\), this represents a portion of the unit circle in the first quadrant (since both \(\cos^2(t)\) and \(\sin^2(t)\) are non-negative). As \(t\) ranges from 0 to \(\frac{\pi}{2}\), the curve moves counterclockwise from point (1,0) to (0,1).

Step 4: Analyzing Curve C4

Curve \(C_4\) has \(x = \ln(t) - t\) and \(y = 1 + t - \ln(t)\), with \(t > 0\). This curve is less straightforward but analyzing points helps, such as \(t = 1\) yields \(x = -1\) and \(y = 1\), and as \(t\) increases, \(x\) keeps decreasing, witnessing more negative \(x\) values due to \( - t\), while \(y\) may increase or decrease. The curve moves in a complex fashion but generally shifts right to left, considering \(x\) becomes negative. Orientation is based on increasing \(t\).

Key Concepts

Curve Sketching

Curve sketching involves drawing the path of a curve given by parametric equations. In this particular exercise, four curves are provided with different parametric forms. Each curve represents a specific path in the xy-plane, driven by distinct mathematical expressions.
Sketching involves identifying:

• Critical points: these help in understanding the curve's trajectory and behaviour.
• The curve's shape: whether it forms a straight line, a parabola, or takes on a more complex shape.
• Orientation: determining the path's direction as a parameter changes.

Breaking down each curve is key to understanding their movements:
- **Straight lines** become apparent with linear parametric relationships, as seen with Curve C1.
- **Parabolic shapes** emerge from quadratic terms, highlighted by Curve C2.
Effective curve sketching benefits from recognizing these patterns and relationships early on.

Parametric Curves

Parametric curves are defined using two separate equations, each dependent on a parameter, often denoted as \(t\). These equations, \(x(t)\) and \(y(t)\), specify the curve by providing coordinates in the xy-plane for each value of \(t\).
For the given curves:

• Curve \(C_1\): Linear relationship between \(x = t\) and \(y = 1 - t\).
• Curve \(C_2\): Interaction between \(x = 1 - t^2\) and \(y = t^2\).
• Curve \(C_3\): A circular relationship using trigonometric identities \(x = \cos^2(t)\), and \(y = \sin^2(t)\).
• Curve \(C_4\): A log-linear combination \(x = \ln(t) - t\) and \(y = 1 + t - \ln(t)\).

The use of parametric equations enables the depiction of more complex curves that aren't functions in the classic sense, since they can form closed loops or multi-directional paths, unlike standard functions that are single-valued.

Orientation of Curves

Orientation refers to the direction that a parametric curve takes as the parameter \(t\) increases. This is one of the unique features in detail accessible through parametric equations. Understanding this direction is crucial for interpreting the curve's path.
Examining the curves:

• **Curve C1**: Moves from top-left to bottom-right as \(t\) increases, which can be directly observed from the linear decrease in \(y\) as \(x\) increases.
• **Curve C2**: Parabolic movement from right to left, originating from the decrease in \(x = 1 - t^2\) as \(t\) varies, due to the symmetry of a parabola centered on the vertical axis.
• **Curve C3**: A counter-clockwise orientation, covering the first quadrant of the unit circle, starting at (1, 0) and ending at (0, 1).
• **Curve C4**: A more complex path that generally trends right to left with increasing pseudolinear changes influenced by logarithmic components, \(x = \ln(t) - t\).

Identifying these orientations aids in predicting where subsequent points will fall, crucial for graphing.

Graphical Analysis

Graphical analysis of parametric curves involves interpreting the graphic data formed by plotting the parametric equations. This not only helps in visualizing the path but also in understanding the broader mathematical phenomena at play.
Considerations in graphical analysis include:

• **Intersections and symmetries**: Check for points where curves cross themselves or others, significant for understanding cyclic or periodic behavior.
• **Behavior at endpoints**: Determine how the curve acts as \(t\) approaches certain values, whether finite or infinite, showing asymptotic behavior or boundedness.
• **Overall shape and pattern**: Recognize repeating patterns due to trigonometric components like in Curve C3, or log-linear transitions as seen in Curve C4.

Through graphical analysis, complex curves are better appreciated for both their aesthetic appeal and the insight they offer into the underlying equations.