1. Graphs of Polynomial Functions
By: Jeffrey Bivin
Lake Zurich High School
Last Updated: October 6, 2009

2. Continuous Function
The graph of a polynomial function f(x) = anxn + an-1xn-1 + an-2xn a1x + a0 is continuous if it has no breaks, holes, vertical asymptotes or sharp turns.

3. Are these graphs continuous?NO NO NO NO
YES
YES
However,
this is not a
polynomial function

4. Leading Coefficient Test & End Behavior
f(x) = anxn + an-1xn-1 + an-2xn a1x + a0
n is even
n is odd
If an is positive
If an is positive
If an is negative
If an is negative

5. Leading Coefficient Test & End Behavior
left (x  )
right (x  )
f(x) = 5x4 + 7x2 + 3 ↑
f(x) = -7x8 + x + 3 ↓
f(x) = 5x5 + 7x2 + 3x
f(x) = -3x5 + 6x3 + 2

6. Real Zeros & Relative Extrema
f(x) = anxn + an-1xn-1 + an-2xn a1x + a0
1. The function has at most n real zeros
2. The graph has at most n-1 relative extrema (relative minima or relative maxima)

7. Finding Real Zeros Terminology F(x) is a polynomial function….
x = a is a zero of the function
x = a is a solution of f(x) = 0
x – a is a factor of f(x)
(a, 0) is an x-intercept of the graph of f(x)

8. Find all real zeros of the polynomial functions
F(x) = x2 – 121
F(x) = 2x2 – 14x + 24
F(x) =
F(x) = x4 – x3 – 20x2
F(x) = (x+2)4(x-3)5(x+4)3(x-5)2

9. 1. F(x) = x2 – 121 = 0 (x – 11)(x + 11) = 0 x – 11 = 0 or x + 11 = 0

10. 2. F(x) = 2x2 – 14x + 24 = 0 X 2(x2 – 7x + 12) = 0 2(x – 4)(x – 3) = 0
2 = 0 or x – 4 = 0 or x – 3 = 0
x = or x = 3

11. 3. X

12. 4. F(x) = x4 – x3 – 20x2 = 0 x2(x2 – x – 20) = 0 x2(x – 5)(x + 4) = 0
x2 = 0 or x – 5 = 0 or x + 4 = 0
x = or x = or x = – 4

13. 5. F(x) = (x+2)4(x-3)5(x+4)3(x-5)2 = 0
x + 2 = 0 or x – 3 = 0 or x + 4 = 0 or x – 5 = 0
x = or x = 3 or x = – 4 or x = 5

14. Graph
F(x) = (x-2)2
F(x) = (x-2)3
F(x) = (x-2)4
F(x) = (x-2)5

15. Multiplicity – repeated zeros
If the multiplicity is odd
 graph crosses x axis
If the multiplicity is even
 graph touches x axis

16. F(x) = (x-5)2(x-3)5(x-1)4(x+4)2(x+7)5
Degree = = 18
(left side): as x  - ∞, y  + ∞
(right side): as x  + ∞, y  + ∞
x = 5 multiplicity 2  touch
x = 3 multiplicity 5  cross
x = 1 multiplicity 4  touch
x = -4 multiplicity 2  touch C T T C T
x = -7 multiplicity 5  cross

17. Find zeros of the polynomial function f(x) = x3 – 4x2 – 25x + 100

18. Find a polynomial function that has the given zeros
f(x) = x3 – 6x2 – x + 30
x = 5
x = 3
x = –2
x – 5 = 0
x – 3 = 0
x + 2 = 0
( x – 5)(x – 3)(x + 2) = 0
(x – 5)(x2 – x – 6) = 0
x3 – x2 – 6x – 5x2 + 5x = 0
x3 – 6x2 – x = 0

19. Find a polynomial function that has the given zeros
f(x) = x3 – 4x2 + 2x + 4

20. Intermediate Value Theorem
Let a and b be real numbers such that a<b. If f(x) is a polynomial function such that f(a) ≠ f(b), then in the interval [a, b], f(x) takes on every value between f(a) and f(b).
What does it mean?
f(b)
f(a) a b

21. How is this especially helpful?
We must have a real zero between the x values of 4 and 5
f(5) = 7
We may have more than one real zero in the interval [4, 5]
f(4) = -3

22. Another Look……. A Table Perspective x y1 -2 9 -1 -5 1 3 2 12 7 4 -6-5 1 3 2 12 7 4 -6
At least one real zero between & -1
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At least one real zero between & 1
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At least one real zero between & 4
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