Functions and Graphs

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Functions and Graphs

• Chapter 2

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Library of Functions Piecewise-defined Functions

• 2.5

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Take Notes on the 10 Basic Functions

• For each function write down
• The name
• The function itself
• The x-intercepts
• The y-intercepts
• All max and min
• The domain
• Where the function is increasing and where
  decreasing
• Draw a sketch of the function.

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10 Basic Functions

• Make flash cards of the basic functions on one
  side an their graphs on the other. There will be
  a quiz where you will have to produce the graph
  given the function and there will be no calculator

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10 Basic Functions

• Know the graphs well enough to be able to state
  the intercepts, the increasing and decreasing
  intervals, the symmetry, and the domain and range.

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10 Basic Functions

• Domain is all reals unless it is the square root
  function.
• Range is all reals except
• Square, square root, absolute value, constant
  function, and greatest integer.

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Piecewise Functions

• Functions that are defined by more than one
  equation.

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Piecewise Functions

• Graph, find the domain, intercepts, and range for
  the piecewise functions on the next slides.

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Piecewise Functions

• Graphing hints are on the next few slides.

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Piecewise Function 1

• 3x if x?0
• F(x)
• 4 if x0

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Piecewise Function 1

• Graph y3x.

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Piecewise Function 1

• Note that it tells you to graph this except when
  x0. So erase the point at which x0 and put an
  open circle on the line at this point.

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Piecewise Function 1

• Now graph y4.

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Piecewise Function 1

• The function says to graph this only if x0. So
  we need to erase all points that do not have x0.
  This ends up as the point (0,4).

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Piecewise Function 2

• x3 if xlt-2
• F(x) -2x-3 if x-2

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Piecewise Function 2

• Graph yx3.

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Piecewise Function 2

• The function wants this line graphed when xlt-2,
  so we need to erase all points on the line that
  have an x value that is greater than or equal to
  -2. We end up with a piece of the line. Since
  we cannot include -2, we put an open circle at
  the point where x-2. We want to include the
  point (-1.999999999, 1.000000001), etc. but we do
  not want to include (-2,1). That is where the
  open circle is useful.

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Piecewise Function 2

• Now we graph
• -2x-3.

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Piecewise Function 2

• The function wants this line graphed when x-2.
  Therefore we want to erase the piece of the line
  that has x values that are lower than -2. Note
  that when x-2, y1 for this part of the
  function. Therefore the hole left by the other
  function gets filled by this one. The final
  result is a closed circle.

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Piecewise Function 3

• 2x5 if -3xlt0
• F(x) -3 if x0
• -5x if xgt0

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Piecewise Function 3

• Graph 2x5.

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Piecewise Function 3

• The function wants this graphed when x is greater
  than or equal to -3 and less than zero. We need
  to erase all points on this line that have x
  values that are equal to or greater than 0 and
  less than -3.

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Piecewise Function 3

• Now we graph y-3.

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Piecewise Function 3

• We want to graph this when x0, so we just need
  the point (0,-3). We can erase the rest of the
  line and just leave this point.

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Piecewise Function 3

• Now graph -5x.

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Piecewise Function 3

• The function says to graph this when x is greater
  than zero. We need to erase all points that have
  x as negative or zero. This will leave a piece
  of this line.

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Piecewise Functions

• Domain is looked at on the next few slides.

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Piecewise Function 1

• 3x if x?0
• F(x)
• 4 if x0

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Piecewise Function 1

• The domain is found by looking at the criteria in
  the right column of the function. In this case
  it has a rule for any x not equal to zero and one
  for x0, therefore any x has a rule, so the
  domain is all reals.

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Piecewise Function 2

• x3 if xlt-2
• F(x) -2x-3 if x-2

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Piecewise Function 2

• In the right hand column here there is a rule for
  xlt-2 and x greater than or equal to -2. The
  union of these results is all reals. In other
  words, if you graph this all on one number line
  you would graph all xs. Domain is all reals
  here.

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Piecewise Function 3

• 2x5 if -3xlt0
• F(x) -3 if x0
• -5x if xgt0

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Piecewise Function 3

• Lets graph the right hand column to find that we
  graph -3 for x, 0 for x, all x greater than zero,
  and xs between -3 and zero. The union would be
  x greater than or equal to -3. Domain is x-3.

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Piecewise Functions

• Intercepts are found by naming all of the
  coordinates that lie on the x and y axis. Do not
  name any open circle points.

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Intercepts are (-3,0) and (0,-3)
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Intercepts are (-5/2,0) and (0,4)
F(x)
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Intercept is (0,4)
F(x)
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Piecewise Functions

• Range is found finding the lowest y graphed and
  the highest y graphed and seeing if ever y in
  between those values is graphed and then writing
  the result in inequality or interval notation.

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Range y1
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Range ylt5.
F(x)
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Range all ys except 0.
F(x)