5.2 evaluating and graphing polynomial functions day 1 goal: evaluate and graph polynomial functions
TRANSCRIPT
5.2 Evaluating and Graphing Polynomial Functions
DAY 1
Goal:
Evaluate and graph polynomial functions.
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
Where an 0 and the exponents are all whole numbers.
A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.
For this polynomial function, an is the leading coefficient,
a 0 is the constant term, and n is the degree.
an 0
an
an leading coefficient
a 0
a0 constant term n
n
degree
descending order of exponents from left to right.
n n – 1
Common Types of Polynomials
Degree Type Standard Form
0 Constant f(x) = a0 (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
Identifying Polynomial Functions
2 4
3
1. ( ) 3 7
2
. ( ) 3x
a f x x x
b f x x
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.
Identifying Polynomial Functions
2 1
2
. ( ) 6 2
. ( ) 0.5 2
c f x x x x
d f x x x
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.
Evaluating a Polynomial Function
One way to evaluate is to use direct substitution.
PLUG IT IN, PLUG IT IN
4 2
4 2
2 8 5 7 3
(3) 2(3) 8(3) 5(3) 7
162 72 15 7
98
Evaluate f x x x x when x
f
Using Synthetic Substitution
One way to evaluate polynomial functions is to usedirect substitution. Another way to evaluate a polynomialis to use synthetic substitution.
Use synthetic division to evaluate
f (x) = 2 x 4 + 8 x
2 + 5 x 7 when x = 3.
Polynomial in standard form
Using Synthetic Substitution
2 x 4 + 0 x
3 + (–8 x 2) + 5 x + (–7)
2 6
6
10
18
35
30 105
98
The value of f (3) is the last number you write,In the bottom right-hand corner.
The value of f (3) is the last number you write,In the bottom right-hand corner.
2 0 –8 5 –7 CoefficientsCoefficients
3
x-value
3 •
SOLUTION
Polynomial instandard form
THEREMAINDER
Evaluate using Direct Substitution and Synthetic Substitution
31( ) , when x = 4
2f x x x
Direct Substitution Synthetic Substitution
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x + 2
LinearFunction
Degree = 1
Max. Zeros: 1
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
QuadraticFunction
Degree = 2
Max. Zeros: 2
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x3 + 4x2 + 2
CubicFunction
Degree = 3
Max. Zeros: 3
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x4 + 4x3 – 2x – 1
QuarticFunction
Degree = 4
Max. Zeros: 4
POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Odd
Leading Coefficient: +
End Behavior:
As x -∞; f(x) -∞
As x +∞; f(x) +∞
f(x) = x3
POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Odd
Leading Coefficient: –
End Behavior:
As x -∞; f(x) +∞
As x +∞; f(x) -∞
f(x) = -x3
POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Even
Leading Coefficient: +
End Behavior:
As x -∞; f(x) +∞
As x +∞; f(x) +∞
f(x) = x4
POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Even
Leading Coefficient: –
End Behavior:
As x -∞; f(x) -∞
As x +∞; f(x) -∞
f(x) = -x4
Graphing Summary Degree -- highest power exponent or # of factors with variable termsNumber 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 factorsX-intercepts –the zeros, found by setting each factor equal to zeroY- intercept – where the graph crosses the y axis, found by evaluating the function for an x-value of 0
End behavior – four casesEven degree, + leading coefficient –both ends point upEven degree, - leading coefficient—both ends point downOdd degree, + leading coefficient—starts down, ends upOdd degree, - leading coefficient—starts up, ends down