Unit 2: Polynomial Functions Graphs of Polynomial Functions 2.2

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Unit 2 Polynomial FunctionsGraphs of
Polynomial Functions2.2
JMerrill 2005 Revised 2008
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Learning Goal

• To find zeros and use transformations to sketch
  graphs of polynomial functions
• To use the Leading Coefficient Test to determine
  end behavior

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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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Polynomial Quick Graphs
f(x)
f(x)

• From yesterdays activity
• f(x) x2 2x

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Look at the root where the graph of f(x) crossed
the x-axis. What was the power of the factor?

1. 3
2. 2
3. 1

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Look at each root where the graph of a
functionwiggled at the x-axis. Were the powers
even or odd?

1. Even
2. Odd

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Look at each root where the graph of a function
was tangent to the x-axis. What was the power of
the factor?

1. 4
2. 3
3. 2
4. 1

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Describe the end behavior of a function if a gt 0
and n is even.

1. Rise left, rise right
2. Fall left, fall right
3. Rise left, fall right
4. Fall left, rise right

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Describe the end behavior of a function if a lt 0
and n is even.

1. Rise left, rise right
2. Fall left, fall right
3. Rise left, fall right
4. Fall left, rise right

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Describe the end behavior of a function if a gt 0
and n is odd.

1. Rise left, rise right
2. Fall left, fall right
3. Rise left, fall right
4. Fall left, rise right

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Describe the end behavior of a function if a lt 0
and n is odd.

1. Rise left, rise right
2. Fall left, fall right
3. Rise left, fall right
4. Fall left, rise right

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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

• Find all zeros of f(x) - 2x4 2x2
• X 0, 1, -1

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Multiplicity (repeated zeros)

• How many roots?

• How many roots?

3 is a double root. It is tangent to the x-axis
3 is a double root. It is tangent to the x-axis
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
(wiggles)

• How many roots?

• How many roots?

Double roots
Double roots (tangent)
5 roots x 0, 0, 1, 3, 3. 0 and 3 are double
roots
3 roots x 2, 2, 2
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Given Roots, Find a Polynomial Function

• There are many correct solutions. Our solutions
  will be based only on the factors of the given
  roots
• Ex Find a polynomial function with roots 2, 3,
  3
• Turn roots into factors f(x) (x 2)(x 3)(x
  3)
• Multiply factors f(x) (x 2)(x2 6x 9)
• Finish multiplying f(x) x3 8x2 21x -18

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You Do

• Find a polynomial with roots ½, 3, 3
• One answer might be f(x) 2x3 11x2 12x 9

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Sketch graph
f(x) (x - 4)(x - 1)(x 2)
Step 1 Find zeros.
Step 2 Mark the zeros on a number line. Step 3
Determine end behavior Step 4 Sketch the graph
Fall left, rise right
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Sketch graph
f(x) -(x-4)(x-1)(x2)
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You Do
f(x) (x1)2(x-2)
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You Do
f(x) - (x-4)3
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Sketch graph.
f(x) (x-2)2(x3)(x2)
roots -3, -2 and 2
Rise left, rise right
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Write an equation.
Roots -3, 2 and 6
Factors (x3), (x-2) and (x-6)
Factored Form f(x) (x3)(x-2)(x-6)
Polynomial Form f(x) (x3)(x2 8x 12)
x3 5x2 12x 36
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Write equation.
Zeros -2, -1, 3 and 5
Factors (x2), (x1), (x-3) and (x-5)
Factored Form f(x) (x 2)(x 1)(x 3)(x
5) Polynomial Form
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Gateway Problem

• Sketch the graph of f(x) x2(x 4)(x 3)3

Roots?
Double root at x 0 Root at x 4 Triple root at
x -3
Degree of polynomial?
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Rise left Rise right
End Behavior?