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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.1.13c
Chapter 12, Problem 12.1.13c

11–14. Working with parametric equations Consider the following parametric equations.
c. Eliminate the parameter to obtain an equation in x and y.
d. Describe the curve.


x=−t+6, y=3t−3; −5≤t≤5 

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Start with the given parametric equations: \(x = -t + 6\) and \(y = 3t - 3\) with the parameter \(t\) in the interval \([-5, 5]\).
Solve the first equation for \(t\): from \(x = -t + 6\), rearrange to get \(t = 6 - x\).
Substitute this expression for \(t\) into the second equation: \(y = 3(6 - x) - 3\).
Simplify the equation for \(y\) to express it solely in terms of \(x\): \(y = 18 - 3x - 3\), which simplifies further to \(y = 15 - 3x\).
Interpret the resulting equation \(y = 15 - 3x\): this is a linear equation representing a straight line. The parameter interval \(-5 \leq t \leq 5\) will restrict the portion of the line that corresponds to the curve.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves and motions.
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Parameterizing Equations

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove t, resulting in a direct relationship between x and y. This often requires solving one equation for t and substituting into the other, yielding a Cartesian equation of the curve.
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Eliminating the Parameter

Analyzing and Describing the Curve

Describing the curve means interpreting the resulting Cartesian equation or parametric form to identify its shape, type (line, circle, etc.), and domain restrictions. Understanding the parameter range helps determine the portion of the curve represented.
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Summary of Curve Sketching
Related Practice
Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


c. The parametric equations x=t, y=t², for t≥0, describe the complete parabola y=x².

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.  


d. The point (3,π/2) lies on the graph of r=3 cos 2θ.  

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Textbook Question

Spiral arc length Consider the spiral r=4θ, for θ≥0.


c. Show that L′(θ)>0. Is L″(θ) positive or negative? Interpret your answer.

Textbook Question

11–14. Working with parametric equations Consider the following parametric equations.

c. Eliminate the parameter to obtain an equation in x and y.

d. Describe the curve.


x=2 t,y=3t−4;−10≤t≤10 

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Textbook Question

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.


c. x = 1 + 3s, y = 4 + 2s and x = 4 - 3t, y = 6 + 4t

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Textbook Question

Regions bounded by a spiral: Let Rₙ be the region bounded by the nth turn and the (n+1)st turn of the spiral r = e⁻ᶿ in the first and second quadrants, for θ ≥ 0 (see figure).

c. Evaluate lim(n→∞) Aₙ₊₁/Aₙ.


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