Graphs of Functions

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Graphs of Functions

• Written by Clinton Henry

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Table of Contents

• Vocabulary
• Finding the Domain and Range of a Function
• Vertical Line Test
• Increasing and Decreasing Functions
• Relative Minimums and Maximums of a Function
• Step Functions
• Even and Odd Functions
• Conclusion

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Vocabulary

• Graph of a function This function is a
  collection of ordered pairs (x, f(x)) such that x
  is in the domain of f(x).

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Finding the Domain and Range of a Function

• When finding the domain and range of a function,
  you have to consider all the values from top to
  bottom and left to right.
• Domain, as we recall, is all the x values of all
  points included within a function.
• Range, as we recall, is all the y values of all
  points included within a function.

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Finding the Domain and Range of a Function (cont.)

• If we are including the points -1 and 1 then we
  will use brackets to declare that they are
  included but if they are not, then we use
  parentheses ().
• Special note If you have or , then
  because you can never reach either of those
  concepts, they will always be surrounded by
  parenthesis ()

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Finding the Domain and Range of a Function (cont.)

• Lets look at an example.

Here we see a graph of the equation
with endpoints of x2 and
x-1. We also see that at the point where x2
that it will be included and where the point
x-1, is not included.
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Finding the Domain and Range of a Function (cont.)

• From our earlier definitions of Domain and Range,
  we can tell the function has a Domain of (-1,2
  because x -1 has an open circle on it and x 2
  has a closed dot on it.
• We can also tell the Range of the function goes
  from -1, at its highest point to -5 at its
  lowest point. The Range can be labeled like so
  -5,-1

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Vertical Line Test

• To determine if a equation is a function, we can
  use a simple tool called the vertical line test.
• This test shows a set of points in a coordinate
  plane is the graph of y as a function of x if and
  only if (iff) no vertical line intersects the
  graph at more than one point.

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Vertical Line Test (cont.)

• Lets look at some examples.

As we look at the equation
, we can see that, if we
draw a vertical line like the dashed one that is
on there already, this equation is definitely not
a function because at several points of x we have
2 different values of y.
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Vertical Line Test (cont.)

• Lets look at another situation.

As we can see by this graph, when we use the
vertical line test (the dashed line or even the y
axis), each line only crosses the equation once,
so the equation is a function.
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Vertical Line Test (cont.)

• Lets look at a third situation.

As we look at this equation, we have to remember
the wording of the vertical line test. The words
of more than one point of intersection. At x0,
we can see that there is an asymptote but it
still does not have more than one point of
intersection, so that means this equation is a
function.
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Increasing and Decreasing Functions

• To tell if a function, or part of a function is
  increasing, decreasing or constant, we have 3
  points to live by.

If a function is increasing, the graph will be
going up from left to right on the graph. If a
function is decreasing, the graph will be going
down from left to right on the graph. If a
function is constant, the graph will be perfectly
horizontal as we follow the graph from left to
right.
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Increasing and Decreasing Functions (cont.)

• Example

As we can see from this function, the line is
decreasing until x-2 where it becomes constant
until x2. At that point, it begins to increase
after x2.
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Relative Minimums and Maximums of a Function

• The concept of relative maximum and minimums can
  be confusing because they can be other points on
  functions of graphs that are not the greatest
  maximum or minimum of the function.

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Relative Minimums and Maximums of a Function
(cont.)

• Here is an example.

In this example, the maximum and minimum of this
graph is and . There are two other
relative maximums and minimums. They are shown
on the graph.
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Step Functions

• These are specialty functions mostly used in
  business to show what cost is based on the whole
  number of items sold.
• The graphs of such function appear quite
  different, as we will show on next screen.

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Step Functions (cont.)
As you can see, this function usually has each
segment with a hole and the other end as a filled
dot. This graph represents a function because
with the combination of holes and dots, a
vertical line test will show only one
intersection as we go from left to right along
the graph.
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Even and Odd Functions

• When it comes to discussing even and odd
  functions, we first need to discuss symmetry as
  to how it applies to functions.
• Therefore, on the next slide, we will show how
  symmetry exists.

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Even and Odd Functions (cont.)

• If we say that a graph of a function has symmetry
  to the y-axis, if we can see that the points (x,
  y) and (-x, y) lie on the same graph.

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Even and Odd Functions (cont.)

• If we say that a graph of a function has symmetry
  to the origin, then we can see that the points
  (x, y) and (-x, -y) lie on the same graph.

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Even and Odd Functions (cont.)

• If we say that a graph of a equation (not
  function because it fails the vertical line test)
  has symmetry to the x-axis, if we can see that
  the points (x, y) and (x, -y) lie on the same
  graph.

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Even and Odd Functions (cont.)

• Now, that we can see how changing the sign looks
  like on symmetrical equations, we can now discuss
  the concept of even and odd functions.

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Even and Odd Functions (cont.)

• To figure out if a function is odd, even or
  neither, we have to know a couple of rules.
• The next couple of slides will show you how this
  is done.

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Even and Odd Functions (cont.)

• To determine if a function is even, we need to do
  the following steps.

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Even and Odd Functions (cont.)

• To determine if a function is odd, we need to do
  the following steps.

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Even and Odd Functions (cont.)

• If neither apply, then the function is neither an
  even or odd function.

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Conclusion

• I hope you enjoyed and learned from this
  presentation. Please feel free to look at your
  home and continue thru the other sections of this
  unit.
• Please click on the screen to end this
  presentation.