1. 5.2 Evaluating and Graphing Polynomial Functions DAY 1
Goal:
Evaluate and graph polynomial functions.

2. n n – 1 f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0 a 0 a0
EVALUATING POLYNOMIAL FUNCTIONS
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.

3. Common Types of Polynomials
Degree
Type
Standard Form
Constant
f(x) = a (f(x) = -5) 1
Linear
f(x) = x (f(x) = 3x) 2
Quadratic
f(x) = x2 + x + a0 3
Cubic
f(x) = x3 + x2 + x + a0 4
Quartic
f(x) = x4 + x3 + x2 + x + a0

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.

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.

6. Evaluating a Polynomial Function
One way to evaluate is to use direct substitution.
PLUG IT IN, PLUG IT IN

7. 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.

8. 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 6 10 35 98
THE
REMAINDER
The value of f (3) is the last number you write,
In the bottom right-hand corner.

9. Evaluate using Direct Substitution and Synthetic Substitution
Direct Substitution Synthetic Substitution

10. Evaluate the Polynomial Function Using Direct Substitution when x = -2

11. Evaluate the Polynomial Function Using Synthetic Substitution

12. POLYNOMIAL FUNCTIONS f(x) = x + 2 Linear Function Degree = 1
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x + 2
Linear
Function
Degree = 1
Max. Zeros: 1

13. POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2 Quadratic Function Degree = 2
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
Quadratic
Function
Degree = 2
Max. Zeros: 2

14. POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2 Cubic Function Degree = 3
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x3 + 4x2 + 2
Cubic
Function
Degree = 3
Max. Zeros: 3

15. POLYNOMIAL FUNCTIONS f(x) = x4 + 4x3 – 2x – 1 Quartic Function
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x4 + 4x3 – 2x – 1
Quartic
Function
Degree = 4
Max. Zeros: 4

16. f(x) = x3 POLYNOMIAL FUNCTIONS Degree: Odd Leading Coefficient: +
END BEHAVIOR
f(x) = x3
Degree: Odd
Leading Coefficient: +
End Behavior:
As x  -∞; f(x)  -∞
As x  +∞; f(x)  +∞

17. POLYNOMIAL FUNCTIONS f(x) = -x3 Degree: Odd Leading Coefficient: –
END BEHAVIOR
f(x) = -x3
Degree: Odd
Leading Coefficient: –
End Behavior:
As x  -∞; f(x)  +∞
As x  +∞; f(x)  -∞

18. POLYNOMIAL FUNCTIONS f(x) = x4 Degree: Even Leading Coefficient: +
END BEHAVIOR
f(x) = x4
Degree: Even
Leading Coefficient: +
End Behavior:
As x  -∞; f(x)  +∞
As x  +∞; f(x)  +∞

19. POLYNOMIAL FUNCTIONS f(x) = -x4 Degree: Even Leading Coefficient: –
END BEHAVIOR
f(x) = -x4
Degree: Even
Leading Coefficient: –
End Behavior:
As x  -∞; f(x)  -∞
As x  +∞; f(x)  -∞

21. End behavior – four cases
Graphing Summary
Degree -- highest power exponent or # of factors with variable terms
Number of turns -- if there are n zeros, then there are n-1 turns
Leading coefficient– coefficient of highest power variable or
the number in front of all the factors
X-intercepts –the zeros, found by setting each factor equal to zero
Y- intercept – where the graph crosses the y axis,
found by evaluating the function for an x-value of 0
End behavior – four cases
Even degree, + leading coefficient –both ends point up
Even degree, - leading coefficient—both ends point down
Odd degree, + leading coefficient—starts down, ends up
Odd degree, - leading coefficient—starts up, ends down

22. 5.2 Continued: Graphing Polynomial Functions
Will use end behavior to analyze the graphs of polynomial functions.

23. End Behavior
Behavior of the graph as x approaches positive infinity (+∞) or negative infinity (-∞)
The expression x→+∞ : as x approaches positive infinity
The expression x→-∞ : as x approaches negative infinity

24. End Behavior of Graphs of Linear Equations
f(x) = x
f(x) = -x
f(x)→+∞ as x→+∞
f(x)→-∞ as x→-∞
f(x)→-∞ as x→+∞
f(x)→+∞ as x→-∞

25. End Behavior of Graphs of Quadratic Equations
f(x) = x²
f(x) = -x²
f(x)→+∞ as x→+∞
f(x)→+∞ as x→-∞
f(x)→-∞ as x→+∞
f(x)→-∞ as x→-∞

26. Investigating Graphs of Polynomial Functions
How does the sign of the leading coefficient affect the behavior of the polynomial function graph as x→+∞?
How is the behavior of a polynomial functions graph as x→+∞ related to its behavior as x→-∞ when the functions degree is odd? When it is even?

27. Graphing Polynomial Functions
f(x)= x³ + x² – 4x – 1 x -3 -2 -1 1 2 3
f(x)

28. Graphing Polynomial Functions
f(x)= -x4 – 2x³ + 2x² + 4x x -3 -2 -1 1 2 3
f(x)