Problem 10 - Precalculus

Question

Graph the parametric equations after eliminating the parameter t. Specify the direction on the curve corresponding to increasing values of \(t\). \(t\) can be any real number. $$x=t^{2}-1, y=t+1$$

Step-by-Step Solution

Answer

The Cartesian equation is \( x = y^2 - 2y \), and the curve progresses from left to right as \( t \) increases.

Step 1: Identify the Parametric Equations

We are given the parametric equations with parameter \( t \): \( x = t^2 - 1 \) and \( y = t + 1 \). Our goal is to find a relationship between \( x \) and \( y \) by eliminating \( t \).

Step 2: Express t in Terms of y

From the equation \( y = t + 1 \), solve for \( t \).\[ t = y - 1 \]

Step 3: Substitute t into the Equation for x

Substitute \( t = y-1 \) into the equation \( x = t^2 - 1 \).\[ x = (y - 1)^2 - 1 \]

Step 4: Simplify the Equation

Expand and simplify the equation for \( x \).\[ x = (y - 1)^2 - 1 = y^2 - 2y + 1 - 1 \] \[ x = y^2 - 2y \]

Step 5: Identify the Cartesian Equation

The equation \( x = y^2 - 2y \) is the Cartesian form of the parametric equations, representing a parabola in the \( xy \)-plane.

Step 6: Determine the Curve's Direction

For increasing values of \( t \), as \( t \) increases (starting from \( -\infty \) to \( +\infty \)), \( y = t + 1 \) also increases. This indicates that the direction along the curve is from left to right in the \( xy \)-plane. As \( t \) increases, \( y \) sweeps from left to right along this parabola.

Key Concepts

Cartesian equations

A Cartesian equation is an equation that describes a curve or a surface in terms of coordinates such as x and y in a two-dimensional space. It eliminates the parameter, allowing for the expression of a relationship within the familiar Cartesian plane where each point has a specific location on the X and Y axes.
When dealing with parametric equations, the process of transforming them into a Cartesian equation involves removing the parameter, usually denoted as \( t \), by finding an expression for \( t \) in terms of one variable and substituting it into the equation of the second variable.
For example, given \( x = t^2 - 1 \) and \( y = t + 1 \), you would express \( t \) in terms of \( y \) (as \( t = y - 1 \)) and substitute it back into the equation \( x = t^2 - 1 \) to eliminate \( t \).

• The result is a new equation, \( x = y^2 - 2y \), which is now dependent only on \( x \) and \( y \), common for use in Cartesian coordinates.
• This approach converts the parametric form into a single equation that defines the curve explicitly in the Cartesian plane.

parabolas

In mathematics, a parabola is a U-shaped curve that can open in any direction. In the context of Cartesian equations, it is often expressed in the form \( y = ax^2 + bx + c \) if the parabola opens upward or downward. Or correspondingly, \( x = ay^2 + by + c \) if it opens to the left or right.
The key characteristic of a parabola is its symmetry and its single peak or vertex. In this exercise, we have the equation \( x = y^2 - 2y \), which describes a parabola that opens in the horizontal direction.

• The vertex of a parabola described by \( x = y^2 - 2y \) can be calculated by completing the square or using calculus to find the turning point.
• The direction in which the parabola opens depends on the coefficient of \( y^2 \). If positive, it opens to the right, and if negative, to the left.
• In this scenario, the parabola opens to the right as \( y \) increases, allowing \( x \) to change positively.

eliminating parameters

Eliminating parameters is a technique used to convert parametric equations into Cartesian equations. This process simplifies the representation of curves by removing parameters such as \( t \). It is a helpful skill that allows for exploring the properties of a curve without reference to the parameter itself.
Here is how parameters are eliminated:

• Express the parameter in terms of one variable: This step involves rewriting the equation for one variable to isolate and solve for \( t \).
• Substitute back into the other equation: Use the expression derived from the first equation to replace \( t \) in the second equation, effectively removing the parameter.
• Result in a single equation: The refined equation, which no longer includes \( t \), provides a direct relationship between the variables \( x \) and \( y \).

In our example, with \( x = t^2 - 1 \) and \( y = t + 1 \), solving for \( t \) from \( y = t + 1 \) gives \( t = y - 1 \). Substituting this into \( x = t^2 - 1 \) yields the Cartesian equation \( x = y^2 - 2y \). This process is essential for simplifying and analyzing complex curves in easy-to-understand forms.