Download Section 5.1 – Polynomial Functions

Section 5.1 – Polynomial Functions
Defn:
Polynomial function
In the form of: 𝑓 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ 𝑎1 𝑥 + 𝑎0 .
The coefficients are real numbers.
The exponents are non-negative integers.
The domain of the function is the set of all real numbers.
Are the following functions polynomials?
𝑓 𝑥 = 5𝑥 + 2𝑥 2 − 6𝑥 3 + 3
yes
ℎ 𝑥 = 2𝑥 3 (4𝑥 5 + 3𝑥)
yes
𝑔 𝑥 = 2𝑥 2 − 4𝑥 + 𝑥 − 2
no
2𝑥 3 + 3
no
𝑘 𝑥 = 5
4𝑥 + 3𝑥
Section 5.1 – Polynomial Functions
Defn:
Degree of a Function
The largest degree of the function represents the degree
of the function.
The zero function (all coefficients and the constant are
zero) does not have a degree.
State the degree of the following polynomial functions
𝑔 𝑥 = 2𝑥 5 − 4𝑥 3 + 𝑥 − 2
𝑓 𝑥 = 5𝑥 + 2𝑥 2 − 6𝑥 3 + 3
3
5
𝑘 𝑥 = 4𝑥 3 + 6𝑥 11 − 𝑥 10 + 𝑥 12
ℎ 𝑥 = 2𝑥 3 (4𝑥 5 + 3𝑥)
12
8
Section 5.1 – Polynomial Functions
Defn:
Power function of Degree n
In the form of: 𝑓 𝑥 = 𝑎𝑥 𝑛 .
The coefficient is a real number.
The exponent is a non-negative integer.
Properties of a Power Function w/ n a Positive EVEN integer
Even function  graph is symmetric with the y-axis.
The domain is the set of all real numbers.
The range is the set of all non-negative real numbers.
The graph always contains the points (0,0), (-1,1), & (1,1).
The graph will flatten out for x values between -1 and 1.
Section 5.1 – Polynomial Functions
Properties of a Power Function w/ n a Positive ODD integer
Odd function  graph is symmetric with the origin.
The domain and range are the set of all real numbers.
The graph always contains the points (0,0), (-1,-1), & (1,1).
The graph will flatten out for x values between -1 and 1.
Section 5.1 – Polynomial Functions
Transformations of Polynomial Functions
𝑓 𝑥 = 𝑥2 + 2
𝑓 𝑥 = (𝑥 − 2)2
2
𝑓 𝑥 = (𝑥 − 2)2 +2
2
2
2
Section 5.1 – Polynomial Functions
Transformations of Polynomial Functions
𝑓 𝑥 = (𝑥 + 1)5
𝑓 𝑥 = −(𝑥 − 4)3 − 3 𝑓 𝑥 = −(𝑥 − 1)2 + 5
5
1
4
-3
1
Section 5.1 – Polynomial Functions
Defn:
Real Zero of a function
If f(r) = 0 and r is a real number, then r is a real zero of the
function.
Equivalent Statements for a Real Zero
r is a real zero of the function.
r is an x-intercept of the graph of the function.
x – r is a factor of the function.
r is a solution to the function f(x) = 0
Section 5.1 – Polynomial Functions
Defn:
Multiplicity
The number of times a factor (m) of a function is repeated
is referred to its multiplicity (zero multiplicity of m).
Zero Multiplicity of an Even Number
The graph of the function touches the x-axis but does not
cross it.
Zero Multiplicity of an Odd Number
The graph of the function crosses the x-axis.
Section 5.1 – Polynomial Functions
Identify the zeros and their multiplicity
𝑓 𝑥 = 𝑥−3 𝑥+2 3
3 is a zero with a multiplicity of 1. Graph crosses the x-axis.
-2 is a zero with a multiplicity of 3. Graph crosses the x-axis.
𝑔 𝑥 =5 𝑥+4 𝑥−7 2
-4 is a zero with a multiplicity of 1. Graph crosses the x-axis.
7 is a zero with a multiplicity of 2. Graph touches the x-axis.
𝑔 𝑥 = 𝑥 + 1 (𝑥 − 4) 𝑥 − 2 2
-1 is a zero with a multiplicity of 1. Graph crosses the x-axis.
4 is a zero with a multiplicity of 1. Graph crosses the x-axis.
2 is a zero with a multiplicity of 2. Graph touches the x-axis.
Section 5.1 – Polynomial Functions
Turning Points
The point where a function changes directions from increasing to
decreasing or from decreasing to increasing.
If a function has a degree of n, then it has at most n – 1 turning points.
If the graph of a polynomial function has t number of turning points,
then the function has at least a degree of t + 1 .
What is the most number of turning points the following
polynomial functions could have?
𝑔 𝑥 = 2𝑥 5 − 4𝑥 3 + 𝑥 − 2
𝑓 𝑥 = 5𝑥 + 2𝑥 2 − 6𝑥 3 + 3
3-1
5-1
2
4
𝑘 𝑥 = 4𝑥 3 + 6𝑥 11 − 𝑥 10 + 𝑥 12
ℎ 𝑥 = 2𝑥 3 (4𝑥 5 + 3𝑥)
8-1
12-1
11
7
Section 5.1 – Polynomial Functions
End Behavior of a Function
If 𝑓 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ 𝑎1 𝑥 + 𝑎0 , then the end
behaviors of the graph will depend on the first term of the
function, 𝑎𝑥 𝑛 .
If 𝑓 𝑥 = 𝑎𝑥 𝑛 and n is even, then both ends will
approach +.
If 𝑓 𝑥 = −𝑎𝑥 𝑛 and n is even, then both ends will
approach –.
If 𝑓 𝑥 = 𝑎𝑥 𝑛 and n is odd,
then as x  – , 𝑓 𝑥  – and as x  , 𝑓 𝑥  .
If 𝑓 𝑥 = −𝑎𝑥 𝑛 and n is odd,
then as x  – , 𝑓 𝑥   and as x  , 𝑓 𝑥  –.
Section 5.1 – Polynomial Functions
End Behavior of a Function
𝑓 𝑥 = −𝑎𝑥 𝑛 and n is even
𝑓 𝑥 = 𝑎𝑥 𝑛 and n is even
𝑓 𝑥 = 𝑎𝑥 𝑛 and n is odd
𝑓 𝑥 = −𝑎𝑥 𝑛 and n is odd
Section 5.1 – Polynomial Functions
State and graph a possible function.
𝑧𝑒𝑟𝑜𝑠: −1, 2 𝑤𝑖𝑡ℎ 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦 2, 4
𝑑𝑒𝑔𝑟𝑒𝑒 4
𝑥 = −1 𝑥 = 2 𝑥 = 4
𝑥+1=0 𝑥−2=0 𝑥−4=0
𝑔 𝑥 = 𝑥 + 1 (𝑥 − 4) 𝑥 − 2
𝑥 = −1
2
𝑥=4
𝑥 + 1 (−1 − 4) −1 − 2
2
4 + 1 (𝑥 − 4) 4 − 2
2
𝑥 + 1 (−)(+)
−𝑥 − 1
+ (𝑥 − 4)(+)
𝑥−4
Line with negative slope
Line with positive slope
𝑥=2
2 + 1 (2 − 4) 𝑥 − 2
2
→
+ (−)(𝑥 − 2)2
→
−(𝑥 − 2)2
Parabola opening down
Section 5.1 – Polynomial Functions
State and graph a possible function.
𝑔 𝑥 = 𝑥 + 1 (𝑥 − 4) 𝑥 − 2
-1
2
4
2