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Chapter 11: Problem 11

Graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve . $$ x(t)=t^{2}+4, \quad y(t)=t^{2}-4 ; \quad-\infty

Short Answer

Expert verified
The rectangular equation is y = x - 8. The curve starts at (4, -4) and moves to the upper-right as t increases.

Step by step solution

01

- Eliminate the Parameter

To convert the parametric equations into a rectangular equation, eliminate the parameter (t). Given: o x(t) = t^2 + 4 o y(t) = t^2 - 4 We notice both equations have t^2.
02

- Express t^2 from x(t)

From the equation x(t) = t^2 + 4,solve for t^2: \(t^2 = x - 4\).
03

- Substitute t^2 into y(t)

Substitute \(t^2\) from the first equation: y(t) = (x - 4) - 4 = x - 8.
04

- Write the Rectangular Equation

The rectangular equation that satisfies both parametric equations is y = x - 8.
05

- Determine the Orientation

To determine the orientation, consider how x and y change with increasing t. As t increases from -鈭� to 鈭�: o x(t) = t^2 + 4 increases from 4 to 鈭� o y(t) = (t^2 - 4) increases from -4 to 鈭�.So, the curve starts at (4, -4) and moves towards the top-right as t increases.
06

- Sketch the Curve

Draw the graph using the function y = x - 8. Indicate the orientation with arrows pointing from (4, -4) to the upper-right as x and y increase.

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Key Concepts

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

graphing parametric equations
Parametric equations define points on a plane using a pair of functions. In our case, the plane curve is defined by:
  • x(t) = t^2 + 4
  • y(t) = t^2 - 4
To start, create a table of values for t, x(t), and y(t). Evaluate the functions for a range of t-values, such as -2, -1, 0, 1, 2.
For example:
  • When t = 0, x(0) = 0 + 4 = 4 and y(0) = 0 - 4 = -4
  • When t = 1, x(1) = 1 + 4 = 5 and y(1) = 1 - 4 = -3
Plot these points on the coordinate system. The x-values and y-values increase as t increases, tracing the path of the curve.
Use arrows to show the curve's direction, which helps to understand the curve's orientation.
rectangular equation conversion
To convert parametric equations to a single rectangular (Cartesian) equation, eliminate the parameter (t).
First, solve one equation for t. From x(t) = t^2 + 4, we rearrange it to isolate t^2: \( t^2 = x - 4 \)
Substitute this expression into y(t) to remove t.
In our case: y = (x - 4) - 4 = x - 8.
This is the rectangular form of the parametric equations. It's a single equation in terms of x and y that describes the same curve, making it easier to plot.
curve orientation
Curve orientation shows how the curve is traced as the parameter t varies. To determine this for our functions, observe how x and y change as t increases:
  • As t increases from -鈭� to 鈭�, x(t) = t^2 + 4 increases from 4 to 鈭�
  • Similarly, y(t) = t^2 - 4 increases from -4 to 鈭�

Hence, the curve starts at (4, -4) when t is 0 and moves towards the upper-right as t increases.
Orientation is shown by arrows on the graph, indicating the direction from (4, -4) to the right and upwards.

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