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Chapter 10: Problem 4

Sketch the curve by eliminating the parameter, and indicate the direction of increasing \(t\) $$x=t-3, y=3 t-7 \quad(0 \leq t \leq 3)$$

Short Answer

Expert verified
The curve is a line: \( y = 3x + 2 \), direction from \( x = -3 \) to \( x = 0 \).

Step by step solution

01

Express t in terms of x

We start by solving for the parameter \( t \) using the equation for \( x \). Rearrange the equation \( x = t - 3 \) to solve for \( t \):\[ t = x + 3 \].
02

Substitute t into the y equation

Use the expression for \( t \) we found in Step 1 and substitute it into the equation for \( y \). Substitute \( t = x + 3 \) into \( y = 3t - 7 \):\[ y = 3(x + 3) - 7 \].
03

Simplify the y equation

Now simplify the expression \( y = 3(x + 3) - 7 \):\[ y = 3x + 9 - 7 \]. This simplifies further to \[ y = 3x + 2 \].
04

Identify the curve and direction

The result, \( y = 3x + 2 \), represents the equation of a straight line. Since \( x = t - 3 \), as \( t \) increases from 0 to 3, \( x \) increases from -3 to 0. Thus, the line is sketched from \( x = -3 \) to \( x = 0 \) showing the direction of increasing \( t \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Eliminating Parameters
Eliminating the parameter in parametric equations means transforming a pair of equations that depend on a third variable鈥攗sually denoted as \( t \)鈥攊nto a single equation that relates \( x \) and \( y \) directly. This process helps us understand the shape or path of the curve without having to consider the parameter explicitly. To eliminate the parameter, follow these steps:
  • Start with the equation for \( x \) in terms of \( t \). In the example, \( x = t - 3 \).
  • Reorder this equation to express \( t \) as a function of \( x \). Here, it becomes \( t = x + 3 \).
  • Substitute this expression for \( t \) into the equation for \( y \). Using \( y = 3t - 7 \), it becomes \( y = 3(x + 3) - 7 \).
  • Simplify the resulting equation. After simplifying, we obtain \( y = 3x + 2 \).
With these steps, you have converted the parametric form to an explicit form, revealing the underlying relationship between \( x \) and \( y \).
Sketching Curves
Once you have the relation between \( x \) and \( y \), you can begin sketching the curve to visualize it better. The equation we derived, \( y = 3x + 2 \), is a linear equation, revealing that the curve is a straight line. This knowledge simplifies the task of sketching the curve because linear equations are straightforward to plot.Here are tips to sketch the curve:
  • Identify key points: For the equation \( y = 3x + 2 \), select values for \( x \), like the endpoints \( x = -3 \) and \( x = 0 \), to calculate corresponding \( y \) values.
  • Plot these points on a coordinate plane. In this exercise, you'll plot (\(-3, -7\)) and (\(0, 2\)).
  • Draw a straight line through these points. Since the line is an extension of the plotted points, this helps represent its path and direction accurately.
This line represents the curve defined by the parameters in your original equations, showcasing how it changes as \( x \) and \( y \) values vary.
Direction of Increasing Parameter
Understanding the direction of increasing parameter \( t \) is crucial in showing how the curve evolves over the range of \( t \) values given in the original problem. In this example, we were given \( 0 \leq t \leq 3 \):
  • From our earlier steps, \( x = t - 3 \), and as \( t \) moves from 0 to 3, \( x \) alters from \(-3\) to \(0\).
  • The curve thus starts at the point \((-3, -7)\) when \( t=0 \) and ends at the point \((0, 2)\) when \( t=3 \).
  • Tracing this direction on the curve is critical when plotting, as it indicates the path traversed and helps with properly labeling the sketch.
The direction helps illustrate the real-world applications of the curve, such as time-based movements or simply understanding progression along the line from start to end. By being aware of this, you can always assess how a parametric equation maps onto its Cartesian counterpart.

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