Section 3.2 Polynomial Functions and Their Graphs

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Section 3.2Polynomial Functions and Their
Graphs
JMerrill 2005 Revised 2008
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What is a polynomial?

• An expression in the form of
• f(x) anxn an-1xn-1 a2x2 a1x ao
• where n is a non-negative integer and a2, a1,
  and a0 are real numbers.
• The function is called a polynomial function of x
  with degree n.
• A polynomial is a monomial or a sum of terms that
  are monomials
• Polynomials can NEVER have a negative exponent or
  a variable in the denominator!
• The term containing the highest power of x is
  called the leading coefficient, and the power of
  x contained in the leading terms is called the
  degree of the polynomial.

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Significant features

• The graphs of polynomial functions are continuous
  (no breaksyou draw the entire graph without
  lifting your pencil). This is opposed to
  discontinuous functions (remember piecewise
  functions?).
• This data is continuous as opposed to discrete.

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Examples of Polynomials
Degree Name Example
0 Constant 5
1 Linear 3x2
2 Quadratic X2 4
3 Cubic X3 3x 1
4 Quartic -3x4 4
5 Quintic X5 5x4 - 7
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Significant features

• The graph of a polynomial function has only
  smooth turns. A function of degree n has at most
  n 1 turns.
• A 2nd degree polynomial has 1 turn
• A 3rd degree polynomial has 2 turns
• A 5th degree polynomial has

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Cubic Parent Function
X Y
-3 -27
-2 -8
-1 -1
0 0
1 1
2 8
3 27
Draw the parent functions on the graphs. f(x)
x3
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Quartic Parent Function
X Y
-3 81
-2 16
-1 1
0 0
1 1
2 16
3 81
Draw the parent functions on the graphs. f(x)
x4
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Graph and Translate
Start with the graph of y x3. Stretch it by a
factor of 2 in the y direction. Translate it 3
units to the right.
X Y
-3 -27
-2 -8
-1 -1
0 0
1 1
2 8
3 27
X Y
0 -54
1 -16
2 -2
3 0
4 2
5 16
6 54
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Graph and Translate
Start with the graph of y x4. Reflect it
across the x-axis. Translate it 2 units down.
X Y
-3 -83
-2 -18
-1 -3
0 -2
1 -3
2 -18
3 -83
X Y
-3 81
-2 16
-1 1
0 0
1 1
2 16
3 81
X Y
-3 -81
-2 -16
-1 -1
0 0
1 -1
2 -16
3 -81
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Max/Min

• A parabola has a maximum or a minimum

• Any other polynomial function has a local max or
  a local min. (extrema)

Local max
min
max
Local min
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Leading Coefficient Test

• As x moves without bound to the left or right,
  the graph of a polynomial function eventually
  rises or falls like this
• In an odd degree polynomial
• If the leading coefficient is positive, the graph
  falls to the left and rises on the right
• If the leading coefficient is negative, the graph
  rises to the left and falls on the right
• In an even degree polynomial
• If the leading coefficient is positive, the graph
  rises on the left and right
• If the leading coefficient is negative, the graph
  falls to the left and right

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End Behavior

• If the leading coefficient of a polynomial
  function is positive, the graph rises to the
  right.

y x3
y x5
y x2
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Finding Zeros of a Function

• If f is a polynomial function and a is a real
  number, the following statements are equivalent
• x a is a zero of the function
• x a is a solution of the polynomial equation
  f(x)0
• (x-a) is a factor of f(x)
• (a,0) is an x-intercept of f

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Example

• Find all zeros of f(x)x3 x2 2x
• Set function 0 0 x3 x2 2x
• Factor 0 x(x2 x 2)
• Factor completely 0 x(x 2)(x 1)
• Set each factor 0, solve 0 x
• 0 x 2 so x 2
• 0 x 1 so x -1

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You Do

• f(x)-2x4 2x2
• Degree of polynomial?
• Even
• End behavior?
• Falls to the left and falls to the right
• Zeros?
• X 0, 1, -1

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Multiplicity (repeated zeros)

• How many roots?

• How many roots?

3 is a double root
3 is a double root
4 roots x 1, 3, 3, 4.
3 roots x 1, 3, 3.
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Roots of Polynomials
Triple root lies flat then crosses axis

• How many roots?

• How many roots?

Double roots
Double roots
5 roots x 0, 0, 1, 3, 3. 0 and 3 are double
roots
3 roots x 2, 2, 2