Unit 3: Polynomials
Unit 6: Polynomial and Power Functions Name: _____________________________
Day 7 Polynomial Functions and Their Graphs Investigation Date: __________________ Hour: _____
Fill out this table. Sketch the graph of this function using the critical points. Verify your results with a graphing calculator.
| | | | | |
|Function |f(x) = ¼ (x – 1)(x + 3) |f(x) = ¼ (x – 1)(x + 3)2 |f(x) = -¼ (x – 1)(x + 3) (x – 3) |f(x) = ¼ (x – 1)(x + 3) 2 (x – 3) |
|Leading Coefficient “a” | | | | |
|Degree | | | | |
|(count factors) | | | | |
|Number of Linear Factors | | | | |
|(same as degree) | | | | |
|End Behavior | | | | |
|(use arrows) |( ___ , ___ ) |( ___ , ___ ) |( ___ , ___ ) |( ___ , ___ ) |
|Number of | | | | |
|Turning Points | | | | |
| | | | | |
|y-intercept | | | | |
|(substitute x = 0) | | | | |
|Number of |Crossing: |Crossing: |Crossing: |Crossing: |
|Real Zeros | | | | |
| |Touching: |Touching: |Touching: |Touching: |
| | | | | |
| |Total: |Total: |Total: |Total: |
| | | | | |
|Zeros | | | | |
| |[pic] |[pic] |[pic] |[pic] |
| | | | | |
|Sketch of | | | | |
|Graph | | | | |
| | | | | |
|put dots on zeros and | | | | |
|y-intercept | | | | |
Source: Claudia Heinrich and Kimberly Meyer-2008 DACTM Conference
| | | | | |
|Function |f(x) = -¼(x – 1)(x + 3) 2 (x – 3) |f(x) = ¼(x + 1)(x – 1)(x + 3)(x – 3) |f(x) = -¼(x – 1) 2(x + 3)2(x - 3) |f(x) = ¼(x + 1)(x - 1)(x + 2)(x - 3)(x + 3) |
|Leading Coefficient | | | | |
| | | | | |
|Degree | | | | |
|Number of Linear | | | | |
|Factors | | | | |
|End Behavior | | | | |
| |( ___ , ___ ) |( ___ , ___ ) |( ___ , ___ ) |( ___ , ___ ) |
|Number of | | | | |
|Turning Points | | | | |
| | | | | |
|y-intercept | | | | |
|Number of |Crossing: |Crossing: |Crossing: |Crossing: |
|Real Zeros | | | | |
| |Touching: |Touching: |Touching: |Touching: |
| | | | | |
| |Total: |Total: |Total: |Total: |
| | | | | |
|Zeros | | | | |
| |[pic] |[pic] |[pic] |[pic] |
| | | | | |
|Sketch of | | | | |
|Graph | | | | |
Conjecture:
The maximum possible number of turning points is _____________________________________ the degree of the polynomial.
one less than, equal to, one greater than
The maximum possible number of zeros of a polynomial is _________________________________________ its degree.
less than, equal to, greater than
Touching zeros have _______________ multiplicity. Crossing zeros have _______________ multiplicity.
even, odd even, odd
Unit 6: Polynomial and Power Functions Name: _____________________________
Day 7 Polynomial Functions and Their Graphs Worksheet Date: __________________ Hour: _____
Summary of Conjectures from the Graphing Calculator Activity:
The maximum possible number of turning points is one less than the degree of the polynomial.
The maximum possible number of zeros of a polynomial is the same as its degree.
The graph of a polynomial will touch, but not cross, the x-axis at zeros of even multiplicity.
The graph of the polynomial will cross the x-axis at zeros of odd multiplicity.
▪
Directions: Complete each of the following WITHOUT a graphing calculator. Sketch a general graph of the function. Be sure to include the x-intercepts and y-intercept on this graph!
1) f(x) = ½ (x – 4)(x + 1)(x + 5) 2) f(x) = –2(x + 2)(x – 5)2 3) f(x) = – (x + 4)(x – 3)(x + 1)2
a. Leading Coefficient: a = _______ a. Leading Coefficient: a = _______ a. Leading Coefficient: a = _______
b. Degree: _______ b. Degree: _______ b. Degree: _______
c. End Behavior ( _____, _____ ) c. End Behavior ( _____, _____ ) c. End Behavior ( _____, _____ )
d. Cross: _____ Touch: _____ Total: _____ d. Cross: _____ Touch: _____ Total: _____ d. Cross: _____ Touch: _____ Total: ë²tt_____ Total: _____ d. Cross: _____ Touch: _____ Total: _____
e. y-intercept: _____ e. y-intercept: _____ e. y-intercept: _____
f. Max number of Turning Points: _______ f. Max number of Turning Points: _______ f. Max number of Turning Points: ______
g. Zeros: ____________________________ g. Zeros: ____________________________ g. Zeros: ____________________________
h. Sketch the graph of the function. h. Sketch the graph of the function. h. Sketch the graph of the function.
4) f(x) = ½ (x – 3)2(x + 4)2 5) f(x) = –(x – 3) 2(x + 1) 2(x – 1) 6) f(x) = ½ (x – 2) 3(x + 3) 2(x + 1)
a. Leading Coefficient: a = _______ a. Leading Coefficient: a = _______ a. Leading Coefficient: a = _______
b. Degree: _______ b. Degree: _______ b. Degree: _______
c. End Behavior ( _____, _____ ) c. End Behavior ( _____, _____ ) c. End Behavior ( _____, _____ )
d. Cross: _____ Touch: _____ Total: _____ d. Cross: _____ Touch: _____ Total: _____ d. Cross: _____ Touch: _____ Total: _____
e. y-intercept: _____ e. y-intercept: _____ e. y-intercept: _____
f. Max number of Turning Points: _______ f. Max number of Turning Points: _______ f. Max number of Turning Points: ______
g. Zeros: ____________________________ g. Zeros: ____________________________ g. Zeros: ____________________________$
h. Sketch the graph of the function. h. Sketch the graph of the function. h. Sketch the graph of the function.
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