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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 11 - Parametric Equations and Polar Coordinates문제 11.6.10
11장, 문제 11.6.10

Parabolas


Exercises 9-16 give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.


x² = 6y

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Identify the form of the given parabola equation. The equation \(x^{2} = 6y\) is in the form \(x^{2} = 4py\), which represents a parabola that opens either upward or downward along the y-axis.
Compare the given equation \(x^{2} = 6y\) with the standard form \(x^{2} = 4py\) to find the value of \(p\). Here, \(4p = 6\), so solve for \(p\) by dividing both sides by 4: \(p = \frac{6}{4} = \frac{3}{2}\).
Determine the focus of the parabola. For the form \(x^{2} = 4py\), the focus is located at \((0, p)\). Using the value of \(p = \frac{3}{2}\), the focus is at \((0, \frac{3}{2})\).
Find the equation of the directrix. The directrix is a horizontal line given by \(y = -p\). Substitute \(p = \frac{3}{2}\) to get the directrix equation \(y = -\frac{3}{2}\).
To sketch the parabola, plot the vertex at the origin \((0,0)\), mark the focus at \((0, \frac{3}{2})\), and draw the directrix line \(y = -\frac{3}{2}\). Then, sketch the parabola opening upward, symmetric about the y-axis, passing through points that satisfy \(x^{2} = 6y\).

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주요 개념

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Standard Form of a Parabola

A parabola can be expressed in a standard form such as x² = 4py or y² = 4px, where p represents the distance from the vertex to the focus (and directrix). Recognizing this form helps identify key features like the vertex, focus, and directrix easily.
추천 영상:
3:40
Circles in Standard Form Example 1

Focus and Directrix of a Parabola

The focus is a fixed point inside the parabola where all reflected rays converge, and the directrix is a fixed line outside the parabola. The parabola is the set of points equidistant from the focus and the directrix, which defines its shape and position.
추천 영상:
5:28
Horizontal Parabolas

Graphing Parabolas

Sketching a parabola involves plotting the vertex, focus, and directrix, then drawing a smooth curve equidistant from the focus and directrix. Understanding the orientation (vertical or horizontal) and scale based on p helps create an accurate graph.
추천 영상:
7:42
Properties of Parabolas
관련 실천
교과서 질문

Circles


Sketch the circles in Exercises 53–56. Give polar coordinates for their centers and identify their radii.


r = −2 cos θ

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교과서 질문

Ellipses


Exercises 25 and 26 give information about the foci and vertices of ellipses centered at the origin of the xy−plane. In each case, find the ellipse's standard−form equation from the given information.


Foci: ( ±√2, 0) Vertices: (±2,0)

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교과서 질문

Parabolas


Exercises 9-16 give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.


x = −3y²

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교과서 질문

Shifting Conic Sections


You may wish to review Section 1.2 before solving Exercises 39-56.


Exercises 53-56 give equations for hyperbolas and tell how many units up or down and to the right or left each hyperbola is to be shifted. Find an equation for the new hyperbola, and find the new center, foci, vertices, and asymptotes.


x²/4 − y²/5 = 1, right 2, up 2

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교과서 질문

Hyperbolas


Exercises 27-34 give equations for hyperbolas. Put each equation in standard form and find the hyperbola's asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch.


8x² − 2y² = 16

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교과서 질문

Polar to Cartesian Equations


Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.


r = 3 cos θ

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