Problem 63 - Precalculus: Functions and Graphs

Question

Graph the given curves on the same coordinate plane, and describe the shape of the resulting figure. $$\begin{array}{lll}C_{1}: x=\tan t, & y=3 \tan t ; & 0 \leq t \leq \pi / 4 \\\C_{2}: x=1+\tan t, & y=3-3 \tan t ; & 0 \leq t \leq \pi / 4 \\\C_{3}: x=\frac{1}{2}+\tan t, & y=\frac{3}{2} ; & 0 \leq t \leq \pi / 4\end{array}$$

Step-by-Step Solution

Answer

The curves form a right triangular region with \( C_2 \) as the hypotenuse.

Step 1: Understanding the Parameterization

First, understand that each curve is parameterized by the variable \( t \) where \( 0 \leq t \leq \frac{\pi}{4} \). The curves are given as parametric equations: \( C_1: x = \tan t, y = 3 \tan t \); \( C_2: x = 1 + \tan t, y = 3 - 3 \tan t \); \( C_3: x = \frac{1}{2} + \tan t, y = \frac{3}{2} \).

Step 2: Sketching Curve 1

For \( C_1 \), both \( x \) and \( y \) are \( \tan t \) scaled by different factors. As \( t \) ranges from 0 to \( \frac{\pi}{4} \), \( x \) and \( y \) increase linearly with rate \( 3 \) times greater for \( y \). This results in a straight line through the origin with a slope of \( 3 \), stopping at \( x = 1, y = 3 \).

Step 3: Sketching Curve 2

For \( C_2 \), as \( t \) ranges from 0 to \( \frac{\pi}{4} \), \( x = 1 + \tan t \) starts from 1 and ends at 2. \( y = 3 - 3 \tan t \) decreases from 3 to 0, creating a downward line segment from point \( (1, 3) \) to \( (2, 0) \).

Step 4: Sketching Curve 3

For \( C_3 \), \( x = \frac{1}{2} + \tan t \) increases from \( \frac{1}{2} \) to \( \frac{3}{2} \). However, \( y \) is constant at \( \frac{3}{2} \). This describes a horizontal line starting from \((\frac{1}{2}, \frac{3}{2})\) to \((\frac{3}{2}, \frac{3}{2})\).

Step 5: Combining the Curves on the Coordinate Plane

Now, place all the curves on the same coordinate plane. \( C_1 \) is a diagonal line \( y = 3x \) from \((0, 0)\) to \((1, 3)\). \( C_2 \) is a downward line from \((1, 3)\) to \((2, 0)\). \( C_3 \) is a horizontal line from \((\frac{1}{2}, \frac{3}{2})\) to \((\frac{3}{2}, \frac{3}{2})\).

Step 6: Analyzing the Shape Formed

Connecting the endpoints of these curves, observe that they form an incomplete triangular region. Specifically, it appears as a right triangle where \( C_2 \) functions as the hypotenuse, with \( C_1 \) and segments of the horizontal axis and vertical line at \( x = 2 \) serving as the legs.

Key Concepts

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by two perpendicular number lines: the horizontal line is called the x-axis, and the vertical line named the y-axis. Together, these axes divide the plane into four sections, known as quadrants. Creating a grid of intersecting lines, the coordinate plane facilitates plotting points where each point is described by an ordered pair \((x, y)\).

For effective graphing, it's important to understand the basic layout and how coordinates function:

• The origin \((0, 0)\) is where the x-axis and y-axis intersect.
• Positive x-values extend to the right, while negative extend to the left.
• Positive y-values move upwards, whereas negative values move downwards.

By understanding the coordinate plane, graphing curves becomes a simpler task as it serves as the backdrop on which the equations of curves can be visualized. Each parametric curve laid out on the plane can be understood in terms of its geometric position and relation to other curves or structures.

Graphing

Graphing on the coordinate plane involves plotting equations to visualize relationships and shapes. In parametric graphing, such as in our exercise, values of a parameter, in this case, \(t\), represent distinct points on a curve.

Steps to graph parametric curves like those in the exercise include:

• Select a range for the parameter, here \(0 \leq t \leq \frac{\pi}{4}\).
• Calculate corresponding \(x\) and \(y\) values from parametric equations, and plot them on the coordinate plane.
• Connect these points smoothly to form continuous curves.

Graphing accurately reflects the behavior of parametric equations; for instance, in our exercise, Curve 1 first as a line segment with positive slope, Curve 2 as a line slanting downward, and Curve 3 as a horizontal line. Graphing these on the same coordinate plane reveals the overall geometric structure they form together.

Right Triangle

The concept of a right triangle is fundamental in geometry, defined by having one 90-degree angle. In our exercise, the combined curves form such a figure on the coordinate plane.

The right triangle here is constructed naturally by:

• One leg as the vertical line from (2, 0) to (2, y)
• Another leg by including sections of the x-axis.
• The hypotenuse, which is the longest side, represented by Curve \(C_2\), showing a diagonal from \((1, 3)\) to \((2, 0)\).

This visualization helps in understanding the spatial relationships between curves; seeing how their intersections and alignments form meaningful geometric shapes such as right triangles.

Parametric Equations

Parametric equations allow us to express curves in terms of a third variable, often time \(t\). Unlike regular equations that directly relate \(x\) and \(y\), parametric equations define both as separate functions of \(t\).

In our exercise, the parametric equations are:

• Curve 1: \(x = \tan t, \ y = 3 \tan t\)
• Curve 2: \(x = 1 + \tan t, \ y = 3 - 3 \tan t\)
• Curve 3: \(x = \frac{1}{2} + \tan t, \ y = \frac{3}{2}\)

By substituting in values for \(t\) within given limits, we obtain specific \(x\) and \(y\) values that plot points on the curve. This flexible system is useful for describing paths and motions that traditional Cartesian equations might not easily represent, such as curves that loop or spiral.