Problem 21 - A Graphical Approach to College Algebra

Question

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=3 t^{2}, y=t+1, \text { for } t \text { in }(-\infty, \infty)$$

Step-by-Step Solution

Answer

The rectangular equation is \( x = 3(y-1)^2 \) with \( y \) in \((-∞, ∞)\).

Step 1: Express t in terms of y

Given the parametric equations, we first solve for the parameter \( t \) in terms of \( y \). From the equation \( y = t + 1 \), we solve for \( t \) to get: \[ t = y - 1 \]

Step 2: Substitute t into the x equation

Now that we have an expression for \( t \), substitute \( t = y - 1 \) into the equation for \( x = 3t^2 \):\[ x = 3(y - 1)^2 \] This gives us the rectangular equation in terms of \( x \) and \( y \).

Step 3: State the interval for y

Since \( t \) can take any real value \((-∞, ∞)\), and given \( y = t + 1 \), \( y \) can also take any real value \((-∞, ∞)\). Therefore, the appropriate interval for \( y \) is \((-∞, ∞)\).

Key Concepts

Rectangular Equation

Rectangular equations are essential for expressing the relationship between variables in the Cartesian coordinate system. In general, parametric equations use a third variable, often denoted as \( t \), to define both \( x \) and \( y \). This setup can sometimes make it cumbersome to analyze the overall shape or behavior of the curve. By converting parametric equations into rectangular equations, you establish a direct relationship between the \( x \) and \( y \) variables directly.

For instance, consider the parametric equations \( x = 3t^2 \) and \( y = t + 1 \). By finding \( t \) in terms of \( y \), that is \( t = y - 1 \), we can eliminate the parameter \( t \) by substituting into the equation for \( x \). This yields \( x = 3(y - 1)^2 \). This equation is now expressed in rectangular form where \( x \) and \( y \) have a direct algebraic relationship.

• Provides a clearer picture of the curve's geometry
• Facilitates easier graphing
• Helps in identifying function properties like symmetry

Intervals in Algebra

Intervals are used in algebra to describe the range of input values, and they are critical for defining the domain over which an equation holds true. When dealing with parametric equations, intervals often describe the range of the parameter \( t \).

For the given problem, \( t \) belongs to the interval \((-\infty, \infty) \), which implies that it can take any real number value. This is important as it influences the values that \( y \) can take. Since \( y = t + 1 \), the interval for \( y \) remains \( (-∞, ∞) \) as well. This unrestricted interval indicates that our curve will extend infinitely in both the positive and negative directions along the y-axis.

• Helps identify the full extent of the graph
• Clarifies any restrictions on the graphing
• Influences the approach to solving and analyzing the problem

Conversion of Variables

Conversion of variables is a technique used to transition from a parametric form to a standard rectangular form. This process often involves expressing the parameter \( t \) in terms of one of the other variables like \( x \) or \( y \). It is a critical skill for solving systems of equations or transforming the form of an equation to ease analysis.

In our example, the conversion starts by isolating \( t \) in terms of \( y \) using \( y = t + 1 \). Solving for \( t \), we find \( t = y -1 \). By substituting this expression into \( x = 3t^2 \), we derive the rectangular equation \( x = 3(y-1)^2 \). This conversion helps us better understand the relationship between \( x \) and \( y \) without the need for parameter \( t \).

• Simplifies the relationship between variables
• Removes unnecessary parameters
• Enables a more straightforward graphical analysis