Zero’s, Multiplicity, and End Behaviors Graphing Polynomials Zero’s, Multiplicity, and End Behaviors

Defn: A Polynomial Function In the form of: 𝒇 𝒙 = 𝒂 𝒏 𝒙 𝒏 + 𝒂 𝒏−𝟏 𝒙 𝒏−𝟏 + ⋯ 𝒂 𝟏 𝒙+ 𝒂 𝟎 has coefficients that are real numbers, exponents that are non-negative integers. and a domain of all real numbers. Are the following functions polynomials? no 𝑓 𝑥 =5𝑥+2 𝑥 2 −6 𝑥 3 +3 yes 𝑔 𝑥 =2 𝑥 2 −4𝑥+ 𝑥 −2 𝑘 𝑥 = 2 𝑥 3 +3 4 𝑥 5 +3𝑥 ℎ 𝑥 =2 𝑥 3 (4 𝑥 5 +3𝑥) yes no

Determine which of the following are polynomial functions Determine which of the following are polynomial functions. If the function is a polynomial, state its degree. A polynomial of degree 4. Not a polynomial because of the square root since the power is NOT an integer Not a polynomial because of the x in the denominator since the power is negative

Graphs of polynomials are smooth and continuous. No gaps or holes, can be drawn without lifting pencil from paper No sharp corners or cusps This IS NOT the graph of a polynomial This IS the graph of a polynomial

Graphs of Polynomial Functions and Nonpolynomial Functions

Comprehensive Graphs A comprehensive graph of a polynomial function will exhibit the following features: 1. all x-intercepts ( real zero’s if any) 2. the y-intercept 3. enough of the graph to reveal the correct end behaviors 4. all turning points, extreme points, and points of inflection 5. the behavior of the graph at the zero’s (WHERE Y = 0) (WHERE X = 0) (DEGREE AND +/- a) (n-1) (MULTIPLICITY)

X-Intercepts

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,0 ) 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

r = -3, -2, -1, 0, 1, 2, 3 factors ( x + 3) ( x -1) ( x + 2) ( x - 2) Use the x-intercepts to write the factors of the function. r = -3, -2, -1, 0, 1, 2, 3 factors ( x + 3) ( x -1) -3 -2 -1 1 2 3 ( x + 2) ( x - 2) ( x + 1) ( x - 3) x

Graph the zero’s of the function f(x) = x( x-5 ) ( x+2) ( x+ 4) ( x – 3)

END BEHAVIORS

END BEHAVIORS degree of the function leading coefficient depend on the degree of the function and the sign of the leading coefficient

Defn: Degree of a Function The largest exponent of the independent variable represents the degree of the function. State the degree of the following polynomial functions 𝑔 𝑥 =2 𝑥 5 −4 𝑥 3 +𝑥−2 𝑓 𝑥 =5𝑥+2 𝑥 2 −6 𝑥 3 +3 5 3 𝑘 𝑥 = 4𝑥 3 +6 𝑥 11 − 𝑥 10 + 𝑥 12 1 ℎ 𝑥 =2 𝑥 3 (4 𝑥 5 +3𝑥) 12 8

What is the leading coefficient ( a ) of each polynomial? Defn: Leading Coefficient ( a ) The coefficient of the term of greatest degree when the function is in standard form. What is the leading coefficient ( a ) of each polynomial? 𝑔 𝑥 =2 𝑥 5 −4 𝑥 3 +𝑥−2 𝑓 𝑥 =5𝑥+2 𝑥 2 −6 𝑥 3 +3 8 𝑘 𝑥 = 4𝑥 3 +6 𝑥 11 − 𝑥 10 + 𝑥 12 1 ℎ 𝑥 =2 𝑥 3 (4 𝑥 5 +3𝑥)

End Behavior of a Function 𝑓 𝑥 =𝑎 𝑥 𝑛 and n is even 𝑓 𝑥 =−𝑎 𝑥 𝑛 and n is even 𝑓 𝑥 =𝑎 𝑥 𝑛 and n is odd 𝑓 𝑥 =−𝑎 𝑥 𝑛 and n is odd

End Behavior Odd degree a > 0 a < 0 Even degree

End Behavior Odd degree a > 0 a < 0 Even degree

End Behavior Odd degree a > 0 a < 0 Even degree

End Behavior Odd degree a > 0 a < 0 Even degree

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.

n =7 Determine the number of turning points for the function 7 – 1 = 6 x (x + 3) (x - 3) f(x)= (x + 1) (x -1) (x - 2) (x + 2) Determine the number of turning points for the function -3 -2 -1 1 2 3 n =7 7 – 1 = 6

Turning Points What is the most number of turning points the following polynomial functions could have? 𝑓 𝑥 =5𝑥+2 𝑥 2 −6 𝑥 3 +3 𝑔 𝑥 =2 𝑥 5 −4 𝑥 3 +𝑥−2 4 3-1 5-1 2 𝑘 𝑥 = 4𝑥 3 +6 𝑥 11 − 𝑥 10 + 𝑥 12 ℎ 𝑥 =2 𝑥 3 (4 𝑥 5 +3𝑥) 8-1 7 11 12-1

MULTIPLICITY OF ZEROS

y = x³(x + 2)4(x − 3)5 Multiplicity How many REAL Roots does the function have? 12 Real Roots y = x³(x + 2)4(x − 3)5 Multiplicity    0 is a root of multiplicity 3, -2 is a root of multiplicity 4, and 3 is a root of multiplicity 5.  

Multiplicity of Zeros The number of times a factor (x-r) of a function is repeated is referred to as the factors multiplicity

Geogebra

Cross EVEN degree ODD degree Bounce “Wiggle”

y = (x + 2)²(x − 1)³ Multiplicity Answer.  −2 is a root of multiplicity 2, and 1 is a root of multiplicity 3.   These are the 5 roots: −2,  −2,  1,  1,  1.

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 wiggles at 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 bounces at 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 bounces at the x-axis.

Example Find the x-intercepts and multiplicity of f(x) = 2(x+2)2(x-3) Solution: x=-2 is a zero of multiplicity 2 or even x=3 is a zero of multiplicity 1 or odd

State and graph a possible function. 𝒈 𝒙 = 𝒙+𝟏 (𝒙−𝟒) 𝒙−𝟐 𝟐 𝟎+𝟏 (𝟎−𝟒) 𝟎−𝟐 𝟐 -1 2 4 4 f(x) +∞ f(x) +∞ +1 -16 -1 2 4 1 2 1 Y = 𝟎+𝟏 (𝟎−𝟒) 𝟎−𝟐 𝟐 ( 1 ) (-4) ( -2)(-2) = -16

Find the zeros (x intercepts) by setting polynomial = 0 and solving. STEP 1 Determine left and right hand behavior by looking at the highest power on x and the sign of that term. Let’s graph: Step 2 Find the zeros (x intercepts) by setting polynomial = 0 and solving. Multiplying out, highest power would be x4 Zeros are: 0, 3, -4 Step 3 Determine multiplicity of zeros. 0 multiplicity 2 (touches) 3 multiplicity 1 (crosses) -4 multiplicity 1 (crosses) Step 4 Find and plot y intercept by putting 0 in for x Determine maximum number of turns in graph by subtracting 1 from the degree. Degree is 4 so maximum number of turns is 3 Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behavior as a guide.

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