Question

Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=2 t+4, y=t-1 $$

Short answer

The sketch is a line: \( x = 2y + 6 \), traced from lower-left to upper-right as \( t \) increases.

Step-by-step solution

Write the Parametric Equations

The given parametric equations are: \( x = 2t + 4 \) and \( y = t - 1 \). Our goal is to eliminate the parameter \( t \) to find a single equation relating \( x \) and \( y \).

Solve for t from y Equation

Start by isolating \( t \) in the equation for \( y \): \( y = t - 1 \). Add 1 to both sides to solve for \( t \): \( t = y + 1 \).

Substitute for t in x Equation

Substitute \( t = y + 1 \) into the equation for \( x = 2t + 4 \). This gives: \( x = 2(y + 1) + 4 \).

Simplify the Equation

Distribute and simplify the equation: \( x = 2y + 2 + 4 = 2y + 6 \). Thus, the Cartesian equation of the curve is \( x = 2y + 6 \).

Identify the Curve Type and Orientation

The equation \( x = 2y + 6 \) describes a straight line. To determine the orientation, observe that as \( t \) increases, \( y = t - 1 \) increases and \( x = 2t + 4 \) increases, indicating that the curve is traced from lower-left to upper-right.

Key concepts

Eliminating Parameters

In a parametric equation, both x and y are expressed in terms of a third variable, often called a parameter, like \( t \). The goal of eliminating the parameter is to derive a singular equation relating \( x \) and \( y \). This converts the parametric form into a more familiar Cartesian form, which can simplify sketching or analyzing the curve.
To eliminate the parameter, you perform the following steps:

• Solve one of the parametric equations for the parameter \( t \).
• Substitute this expression in the other equation.
• Simplify to find a direct relationship between \( x \) and \( y \).

For the provided example, we begin by solving the equation \( y = t - 1 \) for \( t \), which yields \( t = y + 1 \). Substituting this back into \( x = 2t + 4 \) aids in deriving an equation involving only \( x \) and \( y \). With the parameter gone, the complexity of the curve visualization is greatly reduced.

Cartesian Equation

Once the parameter \( t \) is eliminated, the resulting equation solely involves \( x \) and \( y \) and is known as the Cartesian equation. In this exercise, the conversion resulted in the equation \( x = 2y + 6 \).
The Cartesian equation represents the curve in a straightforward form and can be graphed easily on the Cartesian plane.

• It simplifies the analysis of the curve.
• Helps in identifying the type of curve without dealing with the parameter directly.

In the context of this exercise, the Cartesian equation facilitates identifying the curve as a straight line, making it easier to understand and sketch its behavior.

Curve Orientation

Understanding the orientation of a curve helps in visualizing how the curve is traced over time as the parameter changes. Curve orientation essentially shows the direction in which the curve progresses as the parameter increases.
In the case of the parametric equations provided, by observing the changes in \( t \):

• As \( t \) increases, \( y = t - 1 \) also increases.
• At the same time, \( x = 2t + 4 \) increases as well.

This simultaneous increase indicates that the curve moves from the lower-left section towards the upper-right on the Cartesian plane. Understanding orientation is crucial for sketching the curve accurately and predicting how the path is traversed.

Straight Line

A straight line is one of the simplest forms of curves described in mathematics and is characterized by a constant rate of change. This simple form is evident in the Cartesian equation derived, \( x = 2y + 6 \). The characteristics of a straight line include:

• A linear relationship between \( x \) and \( y \).
• No curves or bends, with a uniform direction.
• The equation typically resembles the form \( y = mx + c \), though here rewritten to fit \( x \).

In our example, the line's slope is indicated by the coefficient of \( y \), which is 2, demonstrating how \( x \) changes twice as much as \( y \). Drawing a straight line according to this equation showcases a predictable path with no deviations, demonstrating the fundamental concept of linearity.