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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.R.54d
Chapter 12, Problem 12.R.54d

53–57. Conic sections
d. Make an accurate graph of the curve.
x = 16y²

Verified step by step guidance
1
Recognize the given equation \(x = 16y^{2}\) as a conic section. Since it is quadratic in \(y\) and linear in \(x\), this represents a parabola that opens either to the right or left.
Rewrite the equation in a standard form to identify the vertex and orientation. Here, \(x = 16y^{2}\) can be seen as \(x = 4p y^{2}\), where \(4p = 16\), so \(p = 4\). This means the parabola opens to the right with vertex at the origin \((0,0)\).
Identify the vertex of the parabola at \((0,0)\), since there are no additional terms shifting the graph horizontally or vertically.
Plot the vertex and use the value of \(p=4\) to find the focus at \((p,0) = (4,0)\) and the directrix at \(x = -p = -4\). These help in accurately sketching the parabola.
Draw the parabola opening to the right, symmetric about the \(x\)-axis, passing through points such as \((16,1)\) and \((16,-1)\) (found by substituting \(y=1\) and \(y=-1\) into the equation) to ensure accuracy.

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Key Concepts

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

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They include parabolas, ellipses, circles, and hyperbolas. Understanding the general forms and properties of these curves helps in identifying and graphing them accurately.
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Parabolas as Conic Sections

Parabola and Its Standard Form

A parabola is a conic section defined as the set of points equidistant from a fixed point (focus) and a line (directrix). The equation x = 16y² represents a parabola opening along the x-axis. Recognizing this form aids in determining its orientation and vertex.
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Circles in Standard Form Example 1

Graphing Techniques for Parabolas

Graphing a parabola involves identifying its vertex, axis of symmetry, direction of opening, and key points. For x = 16y², the vertex is at the origin, and the parabola opens rightward. Plotting points by substituting y-values helps create an accurate graph.
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Properties of Parabolas
Related Practice
Textbook Question

General equations for a circle Prove that the equations  

X = a cos t + b sin t,  y = c cos t + d sin t  

where a, b, c, and d are real numbers, describe a circle of radius R provided a² +c² =b² +d² = R² and ab+cd=0. 

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Textbook Question

14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.

The circle x ² + y ² =9, generated clockwise

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Textbook Question

58–59. Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility.

x²/16 - y²/9 = 1; (20/3, -4)

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Textbook Question

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

e. The hyperbola y²/2 - x²/4 = 1 has no x-intercept.

Textbook Question

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

c. The polar coordinates (3, -3π/4) and (-3, π/4) describe the same point in the plane.

Textbook Question

7–8. Parametric curves and tangent lines

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

x = 8cos t + 1, y = 8sin t + 2, for 0 ≤ t ≤ 2π; t = π/3

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