Section 5.1 – Polynomial FunctionsDefn: Polynomial function In the form of: . The coefficients are real numbers. The exponents are non-negative integers. The domain of the function is the set of all real numbers.

𝑓 (𝑥 )=5𝑥+2 𝑥2−6 𝑥3+3 𝑔 (𝑥 )=2 𝑥2−4 𝑥+√𝑥−2

h (𝑥 )=2 𝑥3 (4 𝑥5+3 𝑥)

Are the following functions polynomials?

yes no

yes𝑘 (𝑥 )= 2𝑥3+3

4 𝑥5+3𝑥no

Section 5.1 – Polynomial FunctionsDefn:

Degree of a FunctionThe 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.

𝑓 (𝑥 )=5𝑥+2 𝑥2−6 𝑥3+3 𝑔 (𝑥 )=2 𝑥5−4 𝑥3+𝑥−2

h (𝑥 )=2 𝑥3 (4 𝑥5+3 𝑥)3 5

8𝑘 (𝑥 )=4 𝑥3+6 𝑥11−𝑥10+𝑥12

12

State the degree of the following polynomial functions

Section 5.1 – Polynomial FunctionsDefn: Power function of Degree n In the form of: . 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 graph will flatten out for x values between -1 and 1.

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).

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 graph will flatten out for x values between -1 and 1.

The domain and range are the set of all real numbers.The graph always contains the points (0,0), (-1,-1), & (1,1).

Section 5.1 – Polynomial Functions Transformations of Polynomial Functions

𝑓 (𝑥 )=𝑥2+2

2

𝑓 (𝑥 )=(𝑥−2)2

2

𝑓 (𝑥 )=(𝑥−2)2+2

2

2

Section 5.1 – Polynomial Functions Transformations of Polynomial Functions

𝑓 (𝑥 )=(𝑥+1)5

1

𝑓 (𝑥 )=−(𝑥−4)3−3 𝑓 (𝑥 )=−(𝑥−1)2+5

5

41-3

Section 5.1 – Polynomial FunctionsDefn: Real Zero of a function

r is a real zero of the function. r is an x-intercept of the graph of the function.

Equivalent Statements for a Real Zero

x – r is a factor of the function.r is a solution to the function f(x) = 0

If f(r) = 0 and r is a real number, then r is a real zero of the function.

Section 5.1 – Polynomial FunctionsDefn:

The graph of the function touches the x-axis but does not cross it.

Zero Multiplicity of an Even Number

MultiplicityThe number of times a factor (m) of a function is repeated is referred to its multiplicity (zero multiplicity of m).

The graph of the function crosses the x-axis. Zero Multiplicity of an Odd Number

Section 5.1 – Polynomial Functions

3 is a zero with a multiplicity of𝑓 (𝑥 )=(𝑥−3 ) (𝑥+2 )3Identify the zeros and their multiplicity

3.-2 is a zero with a multiplicity of1. Graph crosses the x-axis.

Graph crosses the x-axis.

-4 is a zero with a multiplicity of𝑔 (𝑥 )=5 (𝑥+4 ) (𝑥−7 )2

2.7 is a zero with a multiplicity of1. Graph crosses the x-axis.

Graph touches the x-axis.

-1 is a zero with a multiplicity of𝑔 (𝑥 )=(𝑥+1 )(𝑥−4) (𝑥−2 )2

1.4 is a zero with a multiplicity of1. Graph crosses the x-axis.

Graph crosses the x-axis.2.2 is a zero with a multiplicity of Graph touches the x-axis.

Section 5.1 – Polynomial Functions

If a function has a degree of n, then it has at most n – 1 turning points.

Turning PointsThe point where a function changes directions from increasing to decreasing or from decreasing to increasing.

If the graph of a polynomial function has t number of turning points, then the function has at least a degree of t + 1 .

𝑓 (𝑥 )=5𝑥+2 𝑥2−6 𝑥3+3 𝑔 (𝑥 )=2 𝑥5−4 𝑥3+𝑥−2

h (𝑥 )=2 𝑥3 (4 𝑥5+3 𝑥)3-1 5-1

8-1𝑘 (𝑥 )=4 𝑥3+6 𝑥11−𝑥10+𝑥12

12-1

What is the most number of turning points the following polynomial functions could have?

2 4

7 11

Section 5.1 – Polynomial Functions

If and n is even, then both ends will approach +.

End Behavior of a FunctionIf , then the end behaviors of the graph will depend on the first term of the function, .

If and n is even, then both ends will approach –.

If and n is odd, then as x – , – and as x , .If and n is odd, then as x – , and as x , –.

Section 5.1 – Polynomial Functions

and n is even End Behavior of a Function

and n is even

and n is odd and n is odd

Section 5.1 – Polynomial Functions State and graph a possible function.

𝑥=−1

Line with negative slope

𝑥=−1 𝑥=2 𝑥=4𝑧𝑒𝑟𝑜𝑠 :−1 ,2 h𝑤𝑖𝑡 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦 2 ,4𝑑𝑒𝑔𝑟𝑒𝑒 4

𝑔 (𝑥 )=(𝑥+1 )(𝑥−4) (𝑥−2 )2𝑥+1=0 𝑥−2=0 𝑥−4=0

(𝑥+1 )(−1−4) (−1−2 )2

(𝑥+1 )(−)¿−𝑥−1

𝑥=4

Line with positive slope

(4+1 )(𝑥−4) (4−2 )2

¿𝑥−4

𝑥=2

Parabola opening down(2+1 )(2−4 )(𝑥−2 )2→¿ → −(𝑥−2)2

Section 5.1 – Polynomial Functions State and graph a possible function.

𝑔 (𝑥 )=(𝑥+1 )(𝑥−4) (𝑥−2 )2

42-1