Download 7.1 Polynomial Functions

7.1 Polynomial Functions
Objectives:
1. Evaluate polynomial functions.
2. Identify general shapes of graphs of
polynomial function.
Polynomials
• Polynomials – a monomial or sum of
monomials
• 3x3 – 2x2 + 6x – 7 is a polynomial in one
variable, since it only contains the variable x.
• Polynomial in One Variable
– A polynomial of degree n in one variable is an
expression of the form
• a0xn + a1xn-1+…+an-1x + an
– where the coefficients a0, a1, a2,…an, represent
real numbers, a0 is not zero, and n represents a
nonnegative integer.
Polynomials
• 4x5 – 4x4 + 3x3 – 2x2 + 4
– a0 = 4, a1 = - 4, a2 = 3, a3 = -2, a4 = 0, a5 = 4
• The degree of a polynomial in one variable is the greatest
exponent of its variable.
• The leading coefficient of a polynomial is the coefficient
of the term with the highest degree.
Find the degree and leading coefficient
– -4x4 + 2x – 7
– Degree = 4, Leading Coefficient = -4
– ½ t7 – ¼ t6 + t5 – t4 + 3t3 – 2t2 + t – 6.
– Degree = 7, Leading Coefficient = ½
Determine Attributes of Polynomials
• x2 + 3x – ½
Degree = 2, Leading Coefficient = 1
• 2y – 4 + 6x3
Not a polynomial in one variable.
• -4h3 + 6h – 7h6 + 2
Rewrite as -7h6 – 4h3 + 6h + 2
Degree = 6, Leading Coefficient = - 7
• z3 – 3/z + 7z2 – 2
Not a polynomial, 3/z is not a monomial
Polynomial Functions
• A polynomial function of degree n can be
described by an equation of the form
P(x) = a0xn + a1xn-1 +…+an-1x + an
where the coefficients a0, a1, a2,…, an represent
real numbers, a0 is not zero, and n represents
a nonnegative integer.
• Examples
f(x) = 4x4 – ½ x3 + x2 – x + 4
n = 4, a0 = 4, a1 = - ½ , a2 = 1, a3 = -1 , a4 = 4
Evaluating Polynomials
• Given p(x) = 3x4 – 2x2 + 7, find p(-3)
p(-3) = 3(-3)4 – 2(-3)2 + 7
p(-3) = 3(81) – 2 (9) + 7
p(-3) = 243 – 18 + 7
p(-3) = 232
Find functional values of variables
• Given f(x) = -3x4 + ½ x3 – 4x2 + x, find f(a)
f(a) = -3(a)4 + ½ (a)3 – 4(a)2 + a
f(a) = -3a4 + ½ a3 – 4a2 + a
• Given p(y) = y3 – 2y, find p(t + 1)
p(t + 1) = (t + 1)3 – 2(t + 1)
p(t + 1) = (t + 1)(t + 1)(t + 1) – 2t – 2
p(t + 1) = (t2 + 2t + 1)(t + 1) – 2t – 2
p(t + 1) = t3 + 2t2 + t + t2 + 2t + 1 – 2t – 2
p(t + 1) = t3 + 3t2 + t – 1
Graphs of polynomial functions
Common Graphs
constant function
degree = 0
cubic function, degree = 3
linear function
degree = 1
Quartic Function
degree = 4
quadratic function
degree = 2
Quintic Function, degree = 5
End Behavior of Graphs
• This is the behavior of the graph as x approaches
+ ∞ or - ∞. (positive and negative infinity)
• If the function is EVEN, degree = 2, 4, etc., the
ends of the graph point the same way either up if
leading coefficient is > 0, or down if leading
coefficient is < 0.
• If the function is ODD, degree = 3, 5, etc., the
ends of the graph point in opposite directions
either down to the left/up to the right if leading
coefficient is > 0 or up on the left/down on the
right if leading coefficient is < 0.
• You can also tell the degree of the graph by
counting how many times the line changes
direction.
End Behavior
Examples
End Behavior?
Even
Leading Coefficient?
Negative
Degree?
2
f ( x)  as
x  
f ( x)  as
x  
f ( x) 
End Behavior? Odd
Leading Coefficient?
Positive
Degree? 5
f ( x)  as
End Behavior? Odd
Leading Coefficient?
Negative
Degree? 3
f ( x)  as
x  
f ( x)  as f ( x)  as
x  
x  
x  
Homework
p. 350, 16-44 even