ELEMENTARY & INTERMEDIATE 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: Intermediate Algebra. This section of material only is downloadable for nominal fee at www.lulu.com/content/378813. See "Tell me more about Keely's GM book".
Parabolas
GOLDen Mathematics - Intermediate Algebra: Section 10.1
Supplemental Sites: MathOL Links - Alg 10.1
Topics of Importance
Graph parabolas manually and on calculator
Recognize reflections, stretches, vertical and horizontal shifts
Find the vertex and axis intercept points algebraically and graphically from
standard and general form
Comments and Cautions
Today is really fun and useful stuff! We are going to
explore the graphs of quadratic functions. These functions all have the same
basic shape called a "parabola". Once we get the basic processes under our
belts, then we will be able to analyze graphs and make connections between
quadratic functions in real-world applications and the points on the
corresponding parabolas. It will be good to get a visual grasp on the algebraic
processes of solving quadratic equations :)
Be sure to explore with your calculator at hand what happens to the basic parabola y=x2 when you put a negative in front, a coefficient other than 1 in front, add or subtract a number from the x2, add or subtract a number to the x as in (x ± #)2. Watch for the effects on the vertex and axis intercept points. Take note of how to algebraically and graphically find the vertex and axis intercept points given the quadratic in standard form y=a(x±h)2±k and in general form y=ax2±bx±c.
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 11.3.)
 | ch 11.3 introduces the graph of a basic parabola along with reflections (flips across the x-axis), vertical shifts (up or down), and horizontal shifts (left or right). In the standard form of the equation of the parabola, y=a(x-h)2-k, be sure that you know that the "a" coefficient gives you information about which way the parabola opens and how wide that opening is, the "k" term is the shift up/down, the "h" term is the shift left/right, and the vertex is the point (h,k). |
| ch 11.3 Given the standard form of the equation of the parabola, y=a(x-h)2-k, be sure that you can find the vertex and all intercept points algebraically (see examples 1-2). You should also be able to verify this information graphically. Practice both approaches! |
| ch 11.3 Given the general form of the equation of the parabola, y=ax2+bx+c, be sure that you can find the vertex and all intercept points algebraically (see example 3). You should also be able to verify this information graphically. Practice both approaches! |
| ch 11.3 pg 755 You should MEMORIZE the x-coordinate of the vertex formula, Vx=-b/(2a). |
Applications of Quadratic Functions
GOLDen Mathematics - Intermediate Algebra: Section 10.2
Supplemental Sites: MathOL Links - Alg 10.2
Topics of Importance
Optimization applications (minimum and maximum values)
Comments and Cautions
Today's lesson focuses on applications of quadratic functions and analyzing
parabolas in terms of real-life problems. The majority of the problems here are
optimization problems meaning that you are trying to optimize a function like
profits in a small business. There is a particularly useful example in my
GOLDen Mathematics: Intermediate Algebra book if you have access to it. It
is also important to read my
Calculator Guide:
Extrema Points and practice finding maximum and minimum points using a
graphing calculator.
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 11.3 cont'd.)
| ch 11.3 covers optimization word problems and applications of quadratic functions (examples 4-6). IMO the text goes too far and enters the realm of math 111. Don't freak out! Take advantage of technology to find the extrema points electronically rather than the text's algebraic approach. Further ... |
| ch 11.3 example 4. You should recognize that the graph of the function is an upside down parabola and to find the "maximum height" you need to find the coordinates of the vertex (maximum point) of the parabola. Use a calculator and the MAXIMUM feature to do this instead of computing algebraically. Recognize that the x-coordinate of the vertex gives the horizontal distance and the y-coordinate gives the height of the shot put. |
| ch 11.3 example 5. Finding the actual quadratic profit function is studied more in a college algebra class. But once you have the profit P(x) quadratic function then you should be able to produce its graph, electronically find the vertex using the minimum/maximum feature of your grapher, and interpret what the x and y coordinates represent (such as y dollars of profit earned from the sales of x unites of product). |
| ch 11.3 top pg 760. You can SKIP the "table method" in this class. You are not expected to be able to produce a table on your calculator in this class. Everything you need to accomplish in this section can be and should be done graphically instead. |
| ch 11.3 example 6. Again, the focus at this level of class (IMO) should not be on translating the word problem into a quadratic equation, but instead electronically analyzing a given quadratic equation to solve an application. SKIP all the "Fitting Quadratic Functions to Data" stuff. SKIP any question where you are using "linear regression" to take data and make an equation out of it. SKIP any question where you are asked if a graph of data points represents a quadratic function or linear or etc. |
Functions - Composite and Inverse
GOLDen Mathematics - Intermediate Algebra: Section 10.3
Supplemental Sites: MathOL Links - Alg 10.3
Topics of Importance
Composition of functions: definition, notation
Evaluate compositions (given equations or list of ordered pairs)
One-to-one functions: definition
Determining if a relation is 1:1 (given a graph or list of ordered pairs)
Inverse functions: definition, notation, graphical connection, determine if f-1
exists, find and verify f-1 algebraically
Evaluate inverse functions (given equations or list of ordered pairs)
Comments and Cautions
This material is an extension of
Functions -
Introduction from an elementary algebra course. Be sure you know all the
terminology and processes covered there before tackling today's material. The
composition of functions involved putting one function inside another, i.e.
taking the function of a function, so it is important to know how to combine
functions in more basic ways (+, -, x, ÷) first. The inverse of a
function means to undo it (as a cube root undoes a cube). Some functions
have inverses, some do not. Only those functions that are "one-to-one" have
inverses. Be sure you know (from elementary algebra) how to determine if a
relation is a function (given a list of ordered pairs, mapping, graph, or
equation) before learning to determine if a relation is 1:1.
Note that a relation is not a function if it has same x different y like {(1,2),(3,-2),(1,5)}. A relation is not one-to-one if it has same y different x like {(1,2),(3,-2),(0,2)}. Note that the vertical line test (VLT) can be used to determine if a graph is a function or not and that the horizontal line test (HLT) can be used to determine if a graph is one-to-one or not.
Notation Caution: f -1(x) means "the inverse of f(x)" which is the
function that "undoes" the function f.
f -1(x) does not mean the
reciprocal of f(x). In other words, f -1(x) ≠ 1/f(x) even though x -1
= 1/x.
Use web.clark.edu/skeely/FILES/PDF/095/checklist_fnsgrfs.pdf as a checklist to be sure that you have learned all that you should!
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 8.3.)
| You are strongly urged to review sections 8.1-8.2 (covered in elementary algebra) prior to working on section 8.3. |
| ch 8.3 Everything in this section is super important! Here we work with general functions and their inverses. This forms a foundation on which we will build in the next chapter with the specific exponential and logarithmic functions. You do not need to know how to use the "table feature" of your calculator, but you should be able to do the compositions algebraically. The data may be given as a set of ordered pairs instead of in table form. I.e., this example could be stated as "Given that f(x) = {(0,-2),(1,2),(2,-1),(3,1),(4,0),(5,0),(6,-3)} and g(x) = {(0,3),(1,5),(2,7),(3,9),(4,11),(5,13),(6,15)}, find (g o f)(3) and (f oog)(3)." Which would be solved as (g o f)(3) = g(f(3)) = g(1) = 5 and (f o g)(3) = f(g(3)) = f(9) = undefined. |
Originally written: 2006-006-15
Last revision:
2008-02-23 10:23 PM