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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 12.1.11c
Chapter 12, Problem 12.1.11c

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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Start with the given parametric equations: \(x = 2t\) and \(y = 3t - 4\) where \(-10 \leq t \leq 10\).
Solve the first equation for the parameter \(t\): from \(x = 2t\), we get \(t = \frac{x}{2}\).
Substitute \(t = \frac{x}{2}\) into the second equation \(y = 3t - 4\) to eliminate the parameter \(t\). This gives \(y = 3 \left( \frac{x}{2} \right) - 4\).
Simplify the expression to get the relationship between \(x\) and \(y\): \(y = \frac{3}{2}x - 4\).
Describe the curve: since the equation is linear in \(x\) and \(y\), the curve is a straight line with slope \(\frac{3}{2}\) and \(y\)-intercept \(-4\). The parameter range \(-10 \leq t \leq 10\) corresponds to \(x\) values between \(-20\) and \(20\).

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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 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

Curve Description from Equations

Describing the curve means interpreting the resulting equation or parametric form to identify its geometric shape, such as a line, circle, or parabola. Understanding the domain of t helps determine the portion of the curve represented.
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Related Practice
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=−t+6, y=3t−3; −5≤t≤5 

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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 on a parabola closest to the focus is the vertex.

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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

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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