Polynomial Functions Exponents- Coefficients- Degree- Leading Coefficient-
Section 5.1 – Polynomial Functions Defn: Polynomial function The coefficients are real numbers....
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Transcript of Section 5.1 – Polynomial Functions Defn: Polynomial function The coefficients are real numbers....
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