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Chapter One
Functions, Graphs, and Models
Section 1.1 introduces
graphing utilities.
- How can you choose an appropriate
viewing window?
- How and when do you use the trace,
zoom, and intersect commands?
- How can you use your grapher to
approximate the solution(s) to an equation?
Section 1.2 introduces the idea of
function.
- There are four
representations of a function. Can you name them?
- What is the definition
of a function? Explain it in your own words.
- How can you determine if a relation
is
or is not a function? Think about this in terms of each of the
four possible representations of functions.
- What is the vertical
line test? When is it used?
- How can you determine the
domain
of a function?
- How can you determine the
range
of a function?
- You will need to understand and be
comfortable using the function notation, f(x).
- You will need to understand the
difference between changing input versus changing
output.
- What is a scatterplot?
- What is a scatterplot used
for?
Section 1.3 concentrates on graphical
representations of functions.
- How can you tell if a function is increasing, decreasing,
or constant?
- How can you tell if a function is concave
up or concave down?
- How do you use
intervals to describe the increasing/decreasing, concave up/concave
down behavior of a function?
- How can you determine the x-intercept
of a graph?
- How can you determine the y-intercept
of a graph?
- What is a root or
zero of a function?
- What is a piecewise
defined function?
- How can you evaluate
a piecewise defined function?
- How can you graph
a piecewise defined function?
- What is the
greatest integer function and what does its
graph look like?
- How can you tell if the graph of a
function is continuous or discontinuous?
- What is a local
extreme value?
- How can you
locate the local extreme value(s) for a function?
- You need to be skeptical
of the graphs produced by your calculator.
- How can knowing the
domain
and range of a function help you determine a good viewing window?
- It is important to know that most results
found on your grapher are approximations and
not exact values.
- What is
mathematical modeling? Describe the process in your own words.
- How can this process be applied to
solving
story problems?
Section 1.4 deals with
transformations of functions.
- How can you use a graph
to determine whether a function is even, odd, or
neither?
- How can you use algebra
to determine whether a function is even, odd, or neither?
The material in this section relies heavily on
your familiarity with the basic functions shown on page 57.
Can you fill in the
table below without using your book or notes?
Fill in the table below for each of the five basic function
types.
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Identity Function |
Absolute Value |
Square |
Cube |
Square Root |
Cube Root |
| Describe the behavior of the function |
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| State the domain |
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| State the range |
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| Sketch a typical graph |
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| Write the general formula |
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| What is unique about this type of function? |
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If you cannot fill in this table without using your book or calculator, you are not familiar
enough with the basic types of functions and will likely have trouble with these
concepts later in the course. Come to the QSC or see your prof for
assistance if needed.
Now for the transformation part of this
section:
- You will need to be able to identify
changes to the output, or "y", value of a
function. You will need to be able to understand how a formula and graph of a function are
affected by adding, subtracting or multiplying the output by a positive
constant.
- Same thing goes for changes to the
input, or "x", value of a
function. You will need to be able to understand how a formula and graph of a function are
affected by adding, subtracting or multiplying the input by a positive
constant.
Spend some time working examples of each
type of transformation. Write formulas. Draw graphs. Think
about how different transformations might affect the domain and range of a
function. The following table might help to organize these concepts from
this section. Try to fill it in without your book or notes.
| |
How is the graph of
y = f(x) affected? |
How is the formula of f(x) affected? |
How might the domain of f(x) be affected? |
How might the range of f(x) be affected? |
| Add a positive constant to the
output |
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| Subtract a positive constant from the
output |
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| Add a positive constant to the
input |
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| Subtract a positive constant from the
input |
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| Multiply the
output by a positive constant |
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| Multiply the
input by a positive constant |
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Multiply the output variable by a
negative constant |
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Multiply the input variable by a
negative constant |
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Section 1.5 addresses
combining functions by the operations of addition,
subtraction, multiplication, and division and also introduces composition of
functions.
- What is the
domain
of the resulting function with each type of operation?
- What is the
range
of the resulting function with each type of operation?
Dealing with
function composition:
- The output of the
second (or inner) function becomes the input for the first (or outer)
function.
- Be careful to be sure that the output of the second (inner)
function is ok to use as input for the first (outer) function.
- You will
need to be able to work with compositions of functions using
formulas, graphs,
and tables.
- You will also need to be able to break a complicated function
into two simpler functions (this is called function "decomposition").
Section 1.6 deals with
inverse
functions.
- You will need to understand the
relationships between a function and its inverse (see Theorem 5 on page 92).
- In order to
find the inverse of a function, you need to rewrite the function so that all of
the paired values remain paired, but the order of the pair is reversed.
That means that the roles of input (x) and output (y) are
reversed.
- Follow
the advice and example on page 98 to find an inverse symbolically. You
will need to be able to do this without looking at your notes/book.
- You
will also need to be able to find an inverse graphically
(symmetry across the
line y = x).
- You will also need to be able to find an
inverse numerically (interchange the roles of values listed in a
table).
- Know the meanings of the following:
One to one
Horizontal line test
- Know what one to one and horizontal line
test have to do with inverses.
Please bring this outline to the QSC tutors or to your professor with any
questions you may be having. We are here to help !