Problem 30 - 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=\tan t, \quad y=1 ; \quad-\pi / 2

Step-by-Step Solution

Answer

The equation is \( y = 1 \), a horizontal line with orientation from left to right.

Step 1: Analyze Given Parametric Equations

We are given the parametric equations for the curve as \( x = \tan t \) and \( y = 1 \). Here, \( y \) is constant, indicating that the curve is a horizontal line where \( y = 1 \) for all \( t \).

Step 2: Determine Range of \( x \) Values

Since \( x = \tan t \), we need to consider the domain of \( t \) which is \( -\pi/2 < t < \pi/2 \). The function \( \tan t \) has vertical asymptotes at \( t = -\pi/2 \) and \( t = \pi/2 \), and it ranges from \( -\infty \) to \( \infty \) within this interval.

Step 3: Find the Equation Relating \( x \) and \( y \)

Given that \( y \) is constant and \( x \) can take any real number value (since \( \tan t \) spans all reals within its domain), the equation is simply \( y = 1 \). This line includes all points where \( y = 1 \) for any real \( x \).

Step 4: Sketch the Graph and Indicate Orientation

The graph of the curve is a horizontal line at \( y = 1 \). The points on this line correspond to all \( x \)-values from negative to positive infinity. The orientation can be described by the direction of \( x \) increasing as \( t \) increases from \( -\pi/2 \) to \( \pi/2 \). Therefore, orientation on the graph is from left to right.

Key Concepts

Horizontal Line

In the context of the given parametric equations, the equation for the horizontal line emerges simply from the fact that the y-coordinate, given by \( y = 1 \), remains constant for all values of parameter \( t \). This constancy means that as \( t \) varies over its specified range, the value of \( y \) doesn't change.
A horizontal line in the Cartesian plane is characterized by having the same y-value, no matter what the x-value is.

• All points on this line share the coordinate \( y = 1 \).
• An example of such a line is described by the equation \( y = c \), where \( c \) is a constant number.

This particular setup simplifies many analyses since the effect of \( t \) only influences the x-axis' position, while any change in \( y \) is absent. The line intersects the y-axis solely at \( y=1 \), purposefully ignoring x-value variations.

Domain of Tangent Function

Understanding the domain of the tangent function is crucial in this problem because it dictates the values that \( x \) can assume. The equation \( x = \tan t \) is pivotal, as the tangent function is defined for all real numbers except where it hits its vertical asymptotes.
Looking at the given interval \( -\pi/2 < t < \pi/2 \), we see that:\

• The tangent function is uninterrupted and continuous in this interval.
• Its value transitions smoothly from \(-\infty\) to \(\infty\).

The domain selected ensures that we don't include these undefined points at \( t = -\pi/2 \) and \( t = \pi/2 \), which are where the vertical asymptotes are. This allows the x-values to travel freely over the real numbers, thanks to the infinite range of the tangent function within its valid domain.

Orientation of Curve

The orientation of a curve in parametric equations refers to the direction in which the curve is traced as the parameter \( t \) increases. In our case, this involves tracking how the line at \( y = 1 \) is plotted on a graph as \( t \) moves from \(-\pi/2\) to \(\pi/2\).
As \( t \) progresses through this range, the behavior of the tangent function is such that:

• The x-value increases, originating from a very large negative number and moving towards a very large positive number.
• This means the curve's orientation on the graph is from left to right.

Understanding this directional flow is key when interpreting the parametric equations and visualizing how the graph ultimately materializes through the chosen space. The orientation becomes particularly clear when recognizing the expansive nature of \( \tan t \) over its valid domain.

Range of x-values

The range of x-values in parametric equations can be understood through the function \( x = \tan t \) and the domain \( -\pi/2 < t < \pi/2 \). This setup directly influences the range of \( x \) as it essentially becomes the range of the tangent function over the permissible interval for \( t \).
Given the behavior of tangent, in this interval, we see:

• As \( t \) approaches \( -\pi/2 \), \( x = \tan t \) approaches \(-\infty\).
• Conversely, as \( t \) approaches \( \pi/2 \), \( x = \tan t \) approaches \(\infty\).

This indicates that \( x \) can take any value between \(-\infty\) and \(\infty\), offering a complete coverage of the real number line. Therefore, while \( y \) remains fixed at 1, this indicates extreme flexibility for all possible horizontal positions along the graph. This further solidifies the line as defined at \( y = 1 \) with an infinite span on the x-axis.