1. Evaluating and Graphing Polynomial Functions
6.2
Evaluating and Graphing Polynomial Functions

2. It is NOT a function if there are negative exponents or variable as
A polynomial function is a function of the form
f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0
a 0 a0
constant term
an  0 an
leading coefficient
descending order of exponents from left to right. n
n – 1 n
degree
Where an  0 and the exponents are all whole numbers.
For this polynomial function, an is the leading coefficient,
a 0 is the constant term, and n is the degree.
A polynomial function is in standard form if its terms are
written in descending order of exponents from left to right.
It is NOT a function if there are
negative exponents or variable as
exponents

3. You are already familiar with some types of polynomial
functions. Here is a summary of common types of polynomial functions.
Degree
Type
Standard Form
f (x) = a
Constant
f (x) = a1x + a 1
Linear 2
Quadratic
f (x) = a 2 x 2 + a 1 x + a 3
Cubic
f (x) = a 3 x 3 + a 2 x 2 + a 1 x + a
f (x) = a4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 4
Quartic

4. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type and leading coefficient.
f (x) = x 2 – 3x4 – 7 1 2
SOLUTION
The function is a polynomial function.
Its standard form is f (x) = – 3x x 2 – 7. 1 2
It has degree 4, so it is a quartic function.
The leading coefficient is – 3.

5. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type and leading coefficient.
f (x) = x x
SOLUTION
The function is not a polynomial function because the term 3 x does not have a variable base and an exponent that is a whole number.

6. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type and leading coefficient.
f (x) = 6x x –1 + x
SOLUTION
The function is not a polynomial function because the term 2x –1 has an exponent that is not a whole number.

7. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type and leading coefficient.
f (x) = – 0.5 x +  x 2 – 2
SOLUTION
The function is a polynomial function.
Its standard form is f (x) =  x2 – 0.5x –
It has degree 2, so it is a quadratic function.
The leading coefficient is .

8. f (x) = x 2 – 3 x 4 – 7 f (x) = x 3 + 3x f (x) = 6x2 + 2 x– 1 + x
Identifying Polynomial Functions
Polynomial function?
f (x) = x 2 – 3 x 4 – 7 1 2
f (x) = x x
f (x) = 6x2 + 2 x– 1 + x
f (x) = – 0.5x +  x2 – 2

9. Using Synthetic Substitution
One way to evaluate polynomial functions is to use direct substitution. Another way to evaluate a polynomial is to use synthetic substitution.
Use synthetic division to evaluate
f (x) = 2 x x x - 7 when x = 3.

10. Using Synthetic Substitution
SOLUTION
2 x x 3 + (–8 x 2) + 5 x + (–7)
Polynomial in
standard form
Polynomial in standard form
2 0 –8 5 –7 3
x-value
3 •
Coefficients
Coefficients 6 18 30
105 2 10 35 6 98
The value of f (3) is the last number you write,
In the bottom right-hand corner.

11. Now use direct substitution:
Use synthetic division to evaluate
f (x) = 2 x x x - 7 when x = 3.
You get the same answer
either way!

12. HOMEWORK (DAY 1)
/4-46 evens

13. If “n” is even, the graph of the polynomial is “U-shaped”
meaning it is parabolic (the higher the degree, the
more curves the graph will have in it).
If “n” is odd, the graph of the polynomial is “snake-like” meaning looks like a snake (the higher the degree, the more curves the graph will have in it).

14. Let’s talk about the Leading Coefficient Test:

15. Leading Coefficient Test
Degree is odd
Degree is even
L.C. > 0
Start high,
End high
L.C. < 0
Start low,
End low
L.C. > 0
Start low,
End high
L.C. < 0
Start high,
End low
L.C. = Leading Coefficient

16. f(x) = x4 + 2x2 – 3x f(x) = -x5 +3x4 – x f(x) = 2x3 – 3x2 + 5
Determine the left and right behavior of the graph of each polynomial function.
f(x) = x4 + 2x2 – 3x
f(x) = -x5 +3x4 – x
f(x) = 2x3 – 3x2 + 5

17. Tell me what you know about the equation…

18. Tell me what you know about the equation…
Page 261 #53

19. Tell me what you know about the equation…
Page 261 #54

20. Tell me what you know about the equation…
-x^4+3x^2+4

21. Fundamental Thm of Algebra Zeros of Polynomial Functions:
The graph of f has at most n real zeros
The “n” deals
with the
highest
exponent!

22. How many zeros do these graphs have????

23. x +  is read as “x approaches positive infinity.”
GRAPHING POLYNOMIAL FUNCTIONS
The end behavior of a polynomial function’s graph is the behavior of the graph as x approaches infinity (+ ) or negative infinity (– ). The expression
x  is read as “x approaches positive infinity.”

24. GRAPHING POLYNOMIAL FUNCTIONS
END BEHAVIOR

25. f(x) Graph f (x) = x 3 + x 2 – 4 x – 1. –3 –7 –2 3 –1 1 2 23 SOLUTION
Graphing Polynomial Functions
Graph f (x) = x 3 + x 2 – 4 x – 1.
SOLUTION
To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. x
f(x) –3 –7 –2 3 –1 1 2 23

26. f (x) Graph f (x) = –x 4 – 2x 3 + 2x 2 + 4x. –3 –21 –2 –1 1 3 2 –16
Graphing Polynomial Functions
Graph f (x) = –x 4 – 2x 3 + 2x x.
SOLUTION
To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. x
f (x) –3
–21 –2 –1 1 3 2
–16
–105

27. Assignment
Day 2
/50-78 evens