Chapter 3: Functions and their Graphs
§3.1 Functions. Do you know the function terminology? (Domain, range,
independent variable,
dependent variable .) Can you distinguish between a function and a non-function?
Can you read function notation and make calculations ? What is f (2x − 1), where
§3.2 The Graph of a Function. This section concerns
visual methods. Can you identify a
graph as a function? What is the Vertical- line Test ? Can you discern the doman
and
range of a function from its graph?
§3.3 Properties of a Function. Even and Off
functions seem fundamental. Question: If f
is even then its graph is symmetric with respect to the
. Can you identify
intervals over which a function is increasing and decreasing? Of course you can.
Maxima
and minima? (Certainly, when the function is a quadratic function , we can
compute the
exact max or min, see §4.1.) The average rate of change of f from x to c is
This is an important concept in Calculus. It will not
appear on this exam, however.
§3.4 Library of Functions. You should know that
graphs of all the standard library functions,
excluding the greatest integer function. These basic functions are transformed
to more
complex function via standard transformations (§3.5). Knowledge of these graphs
is key
to sketching transformed versions.
Piecewise defined functions may appear in the exam in the
form of function evaluation
or a simple graphing example.
§3.5 Graphing Techniques: Transformations.
Horizontal and vertical shifting is easy to
understand, you should know these and their functional forms; for example,
horizontal
shifting has the form y = f (x − h). (If h > 0, we shift the graph of y = f (x)
to the
, while, if h < 0, we shift the graph of y = f
(x) to the 
.
§3.5 You should be able to work with horizontal and
vertical stretching and compressing; in
particular, you should be able to identify these transformations.
Can you reflect a graph with respect to the x-axis? The
y-axis?
§3.6 Mathematical Models . An interesting section,
but we only just touched on it. The only
assigned problems were the ones involving the Demand Equation.
Chapter 4: Polynomial and Rational Functions
§4.1 Quadratic Equations and Models . Ahhh,
quadratic functions! A quadratic function
f (x) = ax2+bx+c can be written in the form f (x) = a(x−h)2+k. We see that f
is
a combination of horizontal and vertical shifting, and of vertical
stretching/compressing
of the library function y = x2. There is a cool little formula for h and that
is h = −b/2a.
The axis of symmetry is where? When a > 0, the graph, a parabola don ’t you
know opens
; if a < 0, the parabola opens
. Finding the y- intercept and x -intercepts,
if there are any, may be useful to graphing the parabola.
§4.2 Polynomial Functions. The key to graphing a
polynomials, at least at this level of play is
(1) Knowing the graphs of the power functions; (2) Finding the zeros of the
polynomial;
(3) identifying the multiplicity of each zero ; (4) and taking note of the End
Behavior.
§4.3 Rational Functions I. What is a rational
function? I’m glad you asked, it is the ratio of
two polynomials . Rational functions may have a domain less then the entire real
number
line, they may also have horizontal, vertical or oblique asymptotes. Be prepared
to find
the domain of a rational function and locate its asymptotes.
§4.4 Rational Functions II: Analyzing Graphs. Hmmm,
the assigned problems seem somewhat
limiting. The treatment of this section will be abbreviated, concentrating on
the
homework assignments.
The test, no doubt, will have some short answers
(true/false, fill in the blank) and some questions
involving computation, algebra and graphing . Good luck, but more importantly,
good knowledge.
I shall attempt to construct a fair test over these topics.
Regards, DPS
