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Chapter 12: Problem 42

Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. The segment of the parabola \(y=2 x^{2}-4,\) where \(-1 \leq x \leq 5\)

Short Answer

Expert verified
Question: Find the parametric equations for the curve defined by \(y = 2x^2 - 4\) and the interval \(-1 \leq x \leq 5\). Answer: The parametric equations for the curve are: 1. \(x(t) = t\) 2. \(y(t) = 2t^2 - 4\) with the interval for parameter t being \(-1 \leq t \leq 5\).

Step by step solution

01

Define the parameter

A parameter is a variable that helps us describe a curve through a set of equations. In this problem, we will use the variable \(t\) as our parameter. Since we are given the interval for the x-values, we can create a simple relationship between \(x\) and \(t\) so the interval for the parameter \(t\) corresponds to the x-values.
02

Relate x and the parameter t

To create a simple relationship, we can set \(t\) equal to \(x\): $$ t = x $$ Now we have the interval for the parameter t as \(-1 \leq t \leq 5\).
03

Find y in terms of t

We have the equation for the curve in terms of x: \(y = 2x^2 - 4\). Since we have related \(t\) to \(x\), we can substitute \(t\) into the equation to find the y-equation in terms of \(t\): $$ y = 2t^2 - 4 $$
04

Write the parametric equations

Now we have both x and y in terms of the parameter \(t\). The parametric equations are: $$ x(t) = t $$ $$ y(t) = 2t^2 - 4 $$ And the interval for the parameter t is \(-1 \leq t \leq 5\).

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