Study Tips

TI Downloads

How To Study with DWFK Precalculus

Chapter 8: Analytic Geometry in Two and Three Dimensions

Most of this chapter is devoted to the analytic geometry of conic sections, a topic which has diminished somewhat in emphasis since the introduction of so many other topics into the precalculus curriculum. Additionally, a few of the more obvious extensions of two-dimensional coordinate geometry to three-dimensional coordinate geometry are gathered together into the last section.

It is hard to strike a happy compromise between "all" and "nothing" when it comes to teaching the conic sections. We have chosen to highlight the geometry of the curves, particularly the reflective properties, which have important applications in the real world. This requires knowing about foci and how to find them, which requires a pretty close look at everything that ties the algebra and the geometry of conics together.

The immediate value of this chapter for a first-year calculus course is rather slight, but the material is mandated for precalculus in some state guidelines.

Section 8.1 Section 8.3 Section 8.5

Section 8.2 Section 8.4 Section 8.6

Section 8.1 Conic Sections and Parabolas

Objectives
You will be able to describe the geometry of a parabola (vertex, focus, directrix) when given its quadratic equation and will be able to construct the quadratic equation when given sufficient geometric information. You will be able to model and solve real-world problems involving parabolas.

Key Ideas

Chord (of a parabola) Focus (of a parabola)

Conic section Nappe of a cone

Degenerate conic section Parabaloid of revolution

Directrix (of a parabola) Parabola

Focal length and focal width Standard form (of a parabola)

Study Tips
The opening discussion about general conic sections is simply to establish the correspondence between second-degree equations in two variables and sections of a double-napped right circular cone, a powerful connection indeed. (As the chapter overview notes, it is the second-degree analogue of the correspondence between two-variable linear equations and lines.)

You might think at first that you already know enough about parabolas, which is why we stress the reflection property. This motivates knowing about the focus, which in turn motivates the rest.

Incidentally, the essential reason for the reflection property is evident in Figure 8.3 (although it falls short of a proof). Extend the dotted line between the directrix and the point on the curve so that it continues toward the upper right corner of the page. If you imagine this line to be a ray of light striking the parabola, you can see that the segment from the point to the focus is the reflection of the segment from the point to the directrix.

Technology Tips
You may want to sketch parabolas and tangent lines and reflected rays and the like on your calculator for demonstration purposes. If you want to highlight a point (like the vertex or the focus), remember that you can plot "square" points as one-point scatter plots:

Top

Section 8.2 Ellipses

Objectives
You will be able to describe the geometry of an ellipse (vertices, foci, major and minor axes) when given its quadratic equation and will be able to construct the quadratic equation when given sufficient geometric information. You will be able to model and solve real-world problems involving ellipses.

Key Ideas

Chord (of an ellipse) Foci (of an ellipse)

Eccentricity (of an ellipse) Major axis

Ellipse Minor axis

Ellipsoid of revolution Vertices (of an ellipse)

Study Tips
As with parabolas in the previous section, the emphasis here is on the applications of the ellipse, specifically to planetary orbits and reflections through the foci. It should only be necessary to work with ellipses centered at (0, 0), as you should already know how to translate the graph to the point (h, k).

An alternate approach to deriving the equations of ellipses is to treat them from the start as the result of horizontal and vertical stretches and shrinks of the unit circle. For example, a horizontal stretch by a factor of a and a vertical stretch by a factor of b turns the unit circle into the ellipse in the usual way (see Section 1.5).

Technology Tips
It is not particularly easy to graph ellipses in function mode, since they are not functions. (The top and bottom can be graphed as separate equations.) On the other hand, it is quite easy to graph them in parametric mode:

The parametric representation reinforces the idea that ellipses are just transformations of the unit circle
(x = cos t, y = sin t).
Top

Section 8.3 Hyperbolas

Objectives
You will be able to describe the geometry of a hyperbola (vertices, foci, transverse and conjugate axes, and asymptotes) when given its quadratic equation and will be able to construct the quadratic equation when given sufficient geometric information. You will be able to model and solve real-world problems involving hyperbolas.

Key Ideas

Chord (of a hyperbola) Hyperbola

Conjugate axis Hyperboloid of revolution

Eccentricity (of a hyperbola) Transverse axis

Foci (of a hyperbola) Vertices (of a hyperbola)

Study Tips
Not counting degenerates, there are two conic sections among the Twelve Basic Functions in Chapter 1: One of them is a parabola, and the other is a hyperbola. (Can you pick it out?) Again, it is the hyperbolic shape and its applications that get the emphasis here. It should only be necessary to work with hyperbolas centered at (0, 0), as you should already know how to translate the graph to the point (h, k).

An alternate approach to deriving the equations of hyperbolas is to treat them from the start as the result of horizontal and vertical stretches and shrinks of the "unit hyperbola." For example, a horizontal stretch by a factor of a and a vertical stretch by a factor of b turns the unit hyperbola into the hyperbola in the usual way (see Section 1.5).

Technology Tips
It is not particularly easy to graph hyperbolas in function mode, since they are not functions. (The top and bottom can be graphed as separate equations.) On the other hand, it is quite easy to graph them in parametric mode, complete with asymptotes:

The parametric representation reinforces the idea that hyperbolas are just transformations of the horizontal unit hyperbola (x = sec t, y = tan t) or the vertical unit hyperbola (x = tan t, y = sec t).
Top

Section 8.4 Translation and Rotation of Axes

Objectives
You will be able to rotate coordinate axes in order to graph conic sections that are neither horizontal nor vertical.

Key Ideas

Cross-product term Rotation formulas

Discriminant Rotation of axes

Invariant under rotation Translation of axes

Study Tips
The topic of rotation of axes is included for the sake of completeness. Although it is a lovely application of coordinate geometry, it has a steep learning curve and a small payoff in terms of practical applications. Consequently, many teachers choose to omit it in favor of more urgent topics.

Note that the rotation formulas at the bottom of page 669 can be written concisely in matrix form:

Technology Tips
Nothing short of a grapher with CAS technology can be of much assistance in this section. Graphs of rotated conic sections can be produced on regular graphers by following the procedure outlined in Example 3, but the effort required to split the equation into two functions leads most people to conclude that it is hardly worth it.

On the other hand, people who have understood these sorts of transformations well enough to teach computers how to do them have gone on to make fortunes designing computer graphics.
Top

Section 8.5 Polar Equations of Conics

Objectives
You will understand the general focus-directrix definition of conic sections and will be able to write equations of conic sections in polar form.

Key Ideas

Directrix (of a general conic) Focus (of a general conic)

Eccentricity (of a general conic) Polar equation of a conic

Focal axis (of a general conic) Vertex (of a general conic)

Study Tips
Like the topic of rotation of axes, this topic has a steep learning curve. You who have not studied Sections 6.4 and 6.5 will obviously be unprepared for this section, and even those who have studied polar graphs will be unlikely to digest the material quickly. Take a careful look at your semester calendar before getting into polar equations of conics; it is a lovely topic with historical significance, but not exactly in the precalculus mainstream. (You would not see this material again until second-year calculus, if, indeed, you see it again at all.)

Technology Tips
You will gain some facility at graphing conic sections in polar form as you work through the examples. Be sure to have them work Exploration 1, which not only shows how the eccentricity affects the graph, but also produces a very creditable sketch of a deer tick.

The eccentricity e appears in several places in the section and in the exercises. Be sure that you do not confuse this with the e on your calculator, which is the e of Section 3.1.

You can produce all five graphs of Exploration 1 with a single function entry by using a list for the eccentricity, as shown below. (The window settings are given in the exploration.)

Top

Section 8.6 Three-Dimensional Cartesian Coordinate Systems

Objectives
You will be able to extend two-dimensional formulas from vectors and coordinate geometry to the corresponding formulas for three-dimensions.

Key Ideas

Center (of a sphere) Radius (of a sphere)

Octants Right-handed coordinate frame

Plane (in Cartesian space) Sphere

Quadric surface Standard unit vectors i, j, k

Study Tips
Even teachers who choose to skip Sections 8.4 and 8.5 might choose to cover this section, which is not very difficult and which serves as a good review of two-dimensional vectors and coordinate geometry. Realistically, however, these are not topics that will be used in a first-year calculus course.

If your class does cover this material, you should keep in mind that these formulas are just extensions of the two-dimensional formulas you have already learned. In some cases, the extensions are so obvious that you might fail to appreciate what the formulas are really saying.

Technology Tips
Computers (and some graphing calculators) will produce 3-dimensional graphs, but that is not an application of technology that we are assuming for the purposes of this book.