COLLEGE ALGEBRA
These brief notes are intended to guide you through the textbook and/or other course readings/materials. As you read the textbook pay particular attention to the "topics of importance" and be sure you know how to accomplish each. The "supplemental sites" may provide additional resources on the internet that supplement the topics. Note: this material is extensively elaborated upon in my optional e-book GOLDen Mathematics: College Algebra. This section of material only is downloadable for a nominal fee at << T.B.A. >>. See "Tell me more about Keely's GM book".
Introduction to Conics
GOLDen Mathematics - College Algebra: Section 6.1
Supplemental Sites:
MathOL Links - Calg
6.1
Topics of Importance
Definition of a "conic section"
Name the 3 conic section curves and know how they are formed by slicing a cone
Comments and Cautions
The conic sections are curves formed when a cone is sliced by a plane (see
your text for pictures). Depending on the angle of the plane in relation to the
cone, the curve formed at their intersection is an ellipse, parabola, or
hyperbola. (A circle is usually also included in this list, but actually a
circle is just a special case of an ellipse.) We will study these static curves
in preparation for calculus III where we will study the motion of these curves
(including in polar form). The conics have lots of real-life applications
including parabolic trajectories, elliptical orbits of the planets, and the
reflective properties of hyperbolas. The supplemental sites (linked above) have
some good websites worth checking out.
The formulas for the conic sections (ch 7) and for sequences and series (ch 8) are including on this Conics, Sequences, and Series Formula Sheet. If your final exam is conducted on-campus, these formulas will be printed on the exam too.
Text Notes (These notes refer to College Algebra 4th ed by Blitzer section 7.1 ... first page only.)
 | ch 7.1 pg 604 introduces conics and their applications. This introduction is too brief. Investigate the supplemental sites for more detailed information. |
Parabolas
GOLDen Mathematics - College Algebra: Section 6.2
Supplemental Sites:
MathOL Links - Calg 6.2
Topics of Importance
Terminology: vertex, axis, focus, directrix, latus rectum
Standard equation of parabola: vertical, horizontal, shifted to a vertex not at
the origin
Equation -> graph (by hand and on calc)
Find equation given graph or information about the graph
Applications of parabolas in science, engineering, architecture
Comments and Cautions
You have studied parabolas before as graphs of quadratic equations, but
their formulas were all solved for y (e.g. y=x2-3x-4 or y=(x-3/2)2-25/4
both of which are parabolas that open up from V(3/2,25/4). Can you complete the
square to transform the first equation to the second? Can you verify this vertex
from both forms of the equation?). In this section we learn a different form of
the equation of a parabola, one that is solved for the square term (e.g. x2=2y-5).
This form actually gives more information and is the form used most often in
calculus. Be sure that you learn how to find the "p" term in the equation and
its effect on the graph of the parabola. This new approach allows us to graph
"sideways parabolas" (ones that open left/right and are non-functions).
Text Notes (These notes refer to College Algebra 4th ed by Blitzer section 7.3.)
| ch 7.3 contains lots of information! It is important to become avidly familiar with this new approach to parabolic functions and their graphs prior to calculus. |
| ch 7.3 see Lecture Notes - Degenerate Conics. |
Circles and Ellipses
Textbook correspondence: some of chapter 2.8 and all of chapter
7.1
GOLDen Mathematics - College Algebra: Section 6.3
Supplemental Sites:
MathOL Links - Calg 6.3
Topics of Importance
Terminology: foci, vertices, major axis, minor axis, eccentricity
Standard equation of ellipse: vertical, horizontal, shifted to a center not at
the origin
Equation -> graph (by hand and on calc)
Find equation given graph or information about the graph
Applications of ellipses in science, engineering, architecture
Comments and Cautions
Ellipses are stretched circles. They are formed based on the placement of
two focal points rather than one center point. Be sure that you learn how to
find the eccentricity from the equation and its effect on the graph of the
ellipse. Also, the area of an ellipse may or may not be covered in your text,
but you are responsible for its formula. Think about what it may be and let's
discuss this in class! Ellipses are rich in scientific history. Fun research
might include Kepler's Laws of Planetary Motion.
Text Notes (These notes refer to College Algebra 4th ed by Blitzer section 2.8 + 7.1.)
| ch 2.8 read "circles" pg 282-286 and try some corresponding problems prior to starting chapter 7.1. |
| ch 2.8 example 6 and ch 7.1 pg 611 cover the process of converting a general form conic to a standard form conic via complete the square. This process will be used throughout chapter 7. Note that the CTS process shown here is slightly different than the CTS process of solving a quadratic equation because a conic's equation may contain both x's and y's that are squared and thus require that you CTS on the x and CTS on the y. |
| ch 2.8 pg 286 shows how to enter a conic equation into a calculator. This is particularly useful for the non-HP calculators whose graphers don't accept non-function equations. |
| ch 7.1 pg 604 Try drawing an ellipse as described in figure 7.2. Explore the effect of changing the foci on the curve. Fun for adults and kids too! |
| ch 7.1 pg 611-612 Applications of conic sections are sprinkled throughout the chapter. Study a wide variety. |
Hyperbolas
GOLDen Mathematics - College Algebra: Section 6.4
Supplemental Sites:
MathOL Links - Calg 6.4
Topics of Importance
Terminology: foci, vertices, transverse axis, conjugate axis, defining
rectangle, asymptote lines, branches
Standard equation of hyperbola: vertical, horizontal, shifted to a center not at
the origin
Equation -> graph (by hand and on calc)
Find equation given graph or information about the graph
Applications of hyperbolas in science, engineering, architecture
Comments and Cautions
A hyperbola has two "branches" and requires a "double cone" (one atop the
other touching at their vertices) to be sliced by a plane to create the two
branches. The curve of a hyperbola looks somewhat like a parabola but it is not
the same curve at all! A hyperbola is bounded by intersecting asymptote lines,
but a parabola is unrestrained. This changes the way the hyperbola curve grows
in subtle but important ways. The reflective properties of the hyperbola are
distinct from those of a parabola (important difference to note!). Be sure also
to note how the definition of the hyperbola differs from the definition of the
ellipse in terms of distance from the foci to points on the curve.
Text Notes (These notes refer to College Algebra 4th ed by Blitzer section 7.2.)
| ch 7.2 The rectangle used as a foundation for drawing the hyperbolas asymptotes and branches is often called the "defining rectangle" in calculus. |
| ch 7.2 example 6 is particularly useful to study as in includes converting from general to standard form via CTS and graphing a conic that has been translated to a non-origin center. |
General and Degenerate Conics
GOLDen Mathematics - College Algebra: Section 6.5
Supplemental Sites:
MathOL Links - Calg 6.5
Topics of Importance
General equation of conics
Converting standard <-> general equation
Degenerate conics: equations, graphs, how to identify
Comments and Cautions
In calculus you will be expected to know how to identify and graph conics
from both the general form, ax2+bx+cy2+dy+e=0, and the
standard form. The key to understanding the relationship between these two forms
of a conic's equation and the graph is the process of converting from one to the
other including via completing the square. Sometimes, depending on the a b c d e
constants, the equation will result in a point, line, or two intersecting lines.
These cases are called the "degenerate conics" and occur when the plane slicing
the double cone does so at particularly interesting places or at specific
angles. These degenerates provide a great connection between the algebra of
conics and the visual experience. Cool stuff!
Text Notes (These notes refer to College Algebra 4th ed by Blitzer section 7.3.)
| Throughout chapter 7 the text works with both the general form of a conic and the standard form of each conic as well as converting from general form to standard form via CTS. But the text neglects to adequately discuss how to idenitify a conic from the general equation, so let's discuss this in class. |
| ch 7.3 pg 640 discusses, far too briefly, the "degenerate conics". Since the text neglects to discuss how to identify a degenerate conic from the general equation, let's discuss this in class. |
Originally written: 2006-09-04
Last revision:
2008-06-02 02:04 AM