Problem 27 - 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. $$x=(t+1)^{3}, \quad y=(t+2)^{2} ; \quad 0 \leq t \leq 2$$

Step-by-Step Solution

Answer

The curve is described by \( y = (\sqrt[3]{x} + 1)^2 \), with increasing \( t \) indicating left to right orientation.

Step 1: Identify the Parametric Equations

The given parametric equations are: \( x = (t+1)^3 \) and \( y = (t+2)^2 \). These represent a set of equations that define a curve \( C \) in the \( xy \)-plane.

Step 2: Express \( t \) in terms of \( x \)

From the equation \( x = (t+1)^3 \), solve for \( t \): \( t = \sqrt[3]{x} - 1 \).

Step 3: Substitute \( t \) into \( y \)'s equation

Take the expression \( t = \sqrt[3]{x} - 1 \) and substitute it into the equation \( y = (t+2)^2 \):\[ y = (\sqrt[3]{x} - 1 + 2)^2 \] which simplifies to \[ y = (\sqrt[3]{x} + 1)^2. \]

Step 4: Simplify the equation

The equation \( y = (\sqrt[3]{x} + 1)^2 \) represents the relationship between \( x \) and \( y \) that defines the curve \( C \).

Step 5: Analyze the Direction and Sketch the Graph

To determine the direction, observe the change in \( t \) from 0 to 2. As \( t \) increases from 0 to 2, \( x \) increases from \( 1 \) to \( 27 \) and \( y \) increases from \( 4 \) to \( 16 \). The orientation of the graph is therefore from left to right.

Key Concepts

Coordinate Transformation

Coordinate transformation is a key aspect when working with parametric equations. When we have parametric equations like \( x = (t+1)^3 \) and \( y = (t+2)^2 \), they define how each parameter \( t \) maps to specific \( x \) and \( y \) coordinates on the curve \( C \). However, to understand the relationship in the familiar \( xy \)-plane, it's often useful to convert these into a single equation in terms of \( x \) and \( y \).

To achieve this, we solve one of the parametric equations for the parameter \( t \). In this case, we used \( x = (t+1)^3 \) to express \( t \) as \( t = \sqrt[3]{x} - 1 \). This expression can then be substituted into the other equation: \( y = (t+2)^2 \).

• Substitute \( t = \sqrt[3]{x} - 1 \) into the \( y \)-equation.
• Simplify the resulting equation to connect \( x \) and \( y \).

This results in \( y = (\sqrt[3]{x} + 1)^2 \), a transformation that gives the relationship between \( x \) and \( y \) for predicted coordinates on the curve.

Graph Sketching

Sketching the graph of a parametric equation involves plotting the curve based on the computed \( xy \)-relationship. In our case, we derived the equation \( y = (\sqrt[3]{x} + 1)^2 \) from the parametric form. This allows for a direct visualization of the curve in the Cartesian \( xy \)-plane.

To accurately sketch this graph:

• Notice that as \( x \) ranges from 1 to 27 (as \( t \) ranges from 0 to 2), \( y \) progresses from 4 to 16.
• This tells us the bounds within which our graph must fit.
• The graph should smoothly increase as both \( x \) and \( y \) values rise, suggesting a continuously rising curve.

Some key points to plot might include those corresponding to \( t = 0, 1, \) and \(2\), providing a structured guide for the overall direction and shape of the graph. These points act as anchor points to define the curve accurately.

Curve Orientation

Understanding curve orientation in parametric equations means determining the direction in which the curve is traced as the parameter \( t \) increases. With our equations, we saw that as \( t \) rises from 0 to 2, both \( x \) and \( y \) coordinates increase steadily.

This pattern of increase reveals the curve's orientation:

• As \( t \) grows, \( x \) moves from 1 to 27.
• Simultaneously, \( y \) moves from 4 to 16.

These changes suggest that the curve traces a path from left to right on the graph, following the upward journey of both values. By marking arrows along the trajectory of the curve, this orientation can be clearly indicated, helping in intuitive understanding of the parametric equations' behavior over the specified interval of \( t \). This knowledge is vital for both analytical studies and practical applications requiring precise curve tracking.