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Chapter 9: Problem 34

Find parametric equations describing the given curve. The line segment from (4,-2) to (2,-1)

Short Answer

Expert verified
The parametric equations for the given line segment are \(x(t) = 4 - 2t\) and \(y(t) = -2 + t\), with \(t \in [0,1]\).

Step by step solution

01

Calculate the Slope

First, identify the slope of the line which runs between the two given points. The slope, m, can be calculated using the formula \[m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\] Where (x_1, y_1) and (x_2, y_2) are the two given points. Therefore, the slope \[m = \frac{-1 - (-2)}{2 - 4} = 1\]
02

Translate into Vector Form

Represent the line segment in vector form. This is done by starting with the first given point, and then adding the product of a parameter, t, and the vector difference between the two given points. For the purposes of a line segment, t belongs to the interval [0,1]. This can be represented in the form of \((x,y) = (x_1,y_1) + t((x_2 - x_1),(y_2 - y_1))\). Therefore, the vector form of this line is \((x,y) = (4,-2) + t((2-4), (-1+2))\).
03

Establish the Parametric Equations

The final step is to establish the parametric equations based on the vector form of the line segment. This is done by setting x = 4 + 2t, and y = -2 + t. Therefore, the parametric equations describing the given curve are \[x(t) = 4 - 2t\] and \[y(t) = -2 + t\]. With t belonging to [0,1].

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Key Concepts

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

Slope Calculation
To find the equation of a line segment, the first step is usually to calculate the slope. The **slope** is a measure of the steepness of a line and helps you understand how the values of y change with respect to x. For a line passing through two points, you can use the simple formula:
  • Identify your coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \)
  • Apply the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  • Calculating this gives you the slope.
For example, in our scenario where the points are \( (4, -2) \) and \( (2, -1) \):
  • Substitute into the slope formula: \[ m = \frac{-1 - (-2)}{2 - 4} = 1 \]
  • So, the slope is 1, indicating a positive increase in y as x increases.
Vector Form
The vector form is key to articulating the movement from one point to another on a line segment. By using a vector form, you express the line segment by capturing both the start point and the direction in which it moves.
  • Identify your initial point \( (x_1, y_1) \)
  • Find the vector direction using \( (x_2 - x_1, y_2 - y_1) \)
  • Combine these into the formula: \((x,y) = (x_1,y_1) + t((x_2 - x_1),(y_2 - y_1))\)
For the points \( (4, -2) \) and \( (2, -1) \), the vector form becomes:
  • Starting from (4, -2), and using the direction \( (-2, 1) \)
  • Construct: \((x,y) = (4,-2) + t(-2, 1)\)
  • Where \( t \) ranges between \([0,1]\), controlling the extent along the segment.
Line Segment
A line segment connects two points directly. It is different from a line, which extends endlessly. When it comes to line segments, establishing parametric equations can precisely describe this part.
  • Based on vector representation, set up parametric equations:
  • The x-component becomes: \[x(t) = 4 - 2t \]
  • The y-component becomes: \[y(t) = -2 + t \]
These equations provide an elegant way to describe our entire line segment because for each value of \( t \) in \([0,1]\), we get a corresponding point on the segment. This means:
  • \( t = 0 \) gives you the starting point \( (4, -2) \)
  • \( t = 1 \) provides the end point \( (2, -1) \)

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Most popular questions from this chapter

Identify the conic section and find each vertex, focus and directrix. $$4 x^{2}+(y+1)^{2}=16$$

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Determine parametric equations for the curves defined by \(x^{2 n}+y^{2 n}=r^{2 n}\) for integers \(n .\) (Hint: Start with \(n=1\) \(x^{2}+y^{2}=r^{2},\) then think of the general equation as \(\left(x^{n}\right)^{2}+\) \(\left(y^{n}\right)^{2}=r^{2 n} .\) Sketch the graphs for \(n=1, n=2\) and \(n=3\) and predict what the curve will look like for large values of \(n\)

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