Graphing Polynomials Zeros Multiplicity and End Behaviors Defn

Graphing Polynomials Zero’s, Multiplicity, and End Behaviors

Defn: A Polynomial Function • has coefficients that are real numbers, • exponents that are non-negative integers. • and a domain of all real numbers. Are the following functions polynomials? yes no

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 sharp corners or cusps This IS the graph of a polynomial No gaps or holes, can be drawn without lifting pencil from paper This IS NOT 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: (WHERE Y = 0) 1. all x-intercepts ( real zero’s if any) 2. the y-intercept (WHERE X = 0) 3. enough of the graph to reveal the (DEGREE AND +/- a) correct end behaviors 4. all turning points, extreme points, (n-1) and points of inflection (MULTIPLICITY) 5. the behavior of the graph at the zero’s

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

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

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

END BEHAVIORS

END BEHAVIORS 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 5 3 1 8 12

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? 8 1

End Behavior of a Function

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

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

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

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

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.

f(x)= x (x + 1) (x - 2) (x + 3) (x - 3) Determine the number of turning points for the function -3 -2 -1 0 n =7 7– 1=6 1 2 3

Turning Points What is the most number of turning points the following polynomial functions could have? 3 -1 2 5 -1 4 12 -1 11 8 -1 7

MULTIPLICITY OF ZEROS

How many REAL Roots does the function have? 12 Real Roots y = x³(x + 4 2) (x − 5 3) 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

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

Identify the zeros and their multiplicity 3 is a zero with a multiplicity of -2 is a zero with a multiplicity of 1. Graph crosses the x-axis. 3. Graph wiggles at the x-axis. -4 is a zero with a multiplicity of 7 is a zero with a multiplicity of 1. Graph crosses the x-axis. 2. Graph bounces at the x-axis. -1 is a zero with a multiplicity of 4 is a zero with a multiplicity of 2 is a zero with a multiplicity of 1. Graph crosses the x-axis. 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. 2 -1 4 4 f(x) +∞ +1 -16 -1 1 2 2 4 1

Let’s graph: STEP 1 Step 2 Determine left and right hand behavior by looking at the Find the zeros (x intercepts) by setting highest power on x and the sign of that term. polynomial = 0 and solving. Multiplying out, highest power would be x 4 Zeros are: 0, 3, -4 Step 3 Determine multiplicity of zeros. Step 4 0 multiplicity 2 (touches) 3 multiplicity 1 (crosses) -4 multiplicity 1 (crosses) 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.

Text VASTNATURE 777 once to 22333 to join the session. Then respond with A, B, C, or D when the poll is active.

Poll. Ev. com/vastnature 777, as long as the poll is active. (? ) VASTNATURE 777 to 22333 to join the session, then they text Poll. Ev. com/vastnature 777 Text VASTNATURE 777 to 22333 once to join