Warm Up

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Warm Up

• State the degree, leading coefficient if it is
  not a polynomial in one variable explain why

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HW
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7-2. Graphing Polynomial Functions

• Objective
• Graph polynomial functions locate their real
  zeros
• Find the maxima minima of polynomial functions

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Graph Polynomial Functions

• To graph a polynomial functions, make a table of
  values to find several points and then connect
  them to make a smooth curve.
• Knowing the end behavior of the graph will assist
  you in completing the sketch of the graph.

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So Lets Graph!
x f(x) x f(x)
-2.5 0.0
-2.0 0.5
-1.5 1.0
-1.0 1.5
-0.5 2.0

• by making a table
  of values
• Even-degree polynomial with a positive leading
  coefficient, so
• How many zeros?

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Completed Table
x f(x) x f(x)
-2.5 8.4 0.0 0
-2.0 0.0 0.5 -2.8
-1.5 -1.3 1.0 -6.0
-1.0 0.0 1.5 -6.6
-0.5 0.9 2.0 0.0
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Location Principle

• In our first example, notice that the values of
  the function before and after each zero are
  different in sign.
• In general, the solutions of a polynomial occur
  somewhere between pairs of x values _at_ which the
  corresponding y values change signs.
• This property for locating zeroes is called the
  Location Principle

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Key Concept!

• Suppose yf(x) represents a polynomial function
  and a and b are two numbers such that f(a)
  lt 0 and f(b) gt 0. Then the function has at least
  one real zero between a and b.

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Key Concept

• If you have a polynomial f(x) and you have two
  points a and b and f(a) is positive and
  f(b) is negative, there is a solution in
  between a and b

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Locate Zeros of a Function

• Determine consecutive values of x between which
  each real zero of the function f(x)x³-5x²3x2
  is located.
• Then draw the graph.
• Note Since f(x) is a third-degree polynomial
  function, it will have either ___, ___, or ___
  zeros.

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f(x)x³-5x²3x2
X f(x)
-2
-1
0
1
2
3
4
5
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f(x)x³-5x²3x2
X f(x)
-2 -32
-1 -7
0 2
1 1
2 -4
3 -7
4 -2
5 17

• gt CHANGE IN SIGNS
• gt CHANGE IN SIGNS
• gt CHANGE IN SIGNS

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From the previous example

• NOTE The changes in sign indicate that there
  are zeros between x-1 and x0, between x1 and
  x2, and between x4 and x5.

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Maximum Minimum
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HW Page 356 (13-18) a b ONLY Sketch graph
MAKE A TABLE!!!!

• 13-16

• 17-18

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Relative Maximum Minimum

• Point A on graph is relative maximum of the cubic
  function since no other nearby points have a
  greater y-coordinate.
• Likewise, point B is a relative minimum since no
  other nearby points have a lesser y-coordinate.
• These points are often referred to as turning
  points

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Key Points

• The graph of a polynomial function of degree n
  has at most n-1 turning points.
• The plurals of maximum minimum are maxima
  minima

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One more example!
X f(x)
-2
-1
0
1
2
3

• Graph f(x)x³-3x²5. Estimate the
  x-coordinates at which the relative maxima
  relative minima occur.

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Answer

• The values of f(x) change signs between x-2 and
  x-1, indicating a zero of the function.
• The value of f(x) at x0 is greater than the
  surrounding points, so it is a relative maximum.
• The value of f(x) at x2 is less than the
  surrounding points, so it is a relative minimum.

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HW

• Page 356 (13-18) You will need to sketch the
  graph of polynomial functions and may want to use
  graph paper.
• If you Google printable graph paper several
  sites will come up that allow you to print graph
  paper out.

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CALCULATOR!!!

• You can use your graphing calculator to find the
  coordinates of relative maxima and relative
  minima. Enter the polynomial function in the Y
  list and graph the function. Make sure that all
  the turning points are visible in the viewing
  window. Find the coordinates of the minimum and
  maximum points respectively

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Graphing Calculator (Maxima Minima)

• Step 1 Graph the function so that the vertex of
  the parabola is visible.
• Step 2 Select 3minimum or 4maximum from the
  CALC menu.
• Step 3 Using the arrow keys, locate a left bound
  and press ENTER.
• Step 4 Locate a right bound and press ENTER
  twice. The cursor appears on the maximum or
  minimum point of the function. The maximum or
  minimum value is the y-coordinate of that point.