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5.1 Polynomial Functions Target: I can classify polynomials and recognize their leading coefficients and describe end behavior

The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial functions where the degree is n and a is the coefficient look like this: f(x) = anxn + an-1xn-1 + … +a1x + a0 Example: f(x) = 4x4 – 3x3 + 12x2 – 7x + 2 (degree 4)

Degree The degree of the function tells the maximum number of real zeros the function has, or the number of times the graph of the function crosses the x-axis (ex: degree 4 function means there are at most 4 real zeros) f(x) = 4x4 – 3x3 + 12x2 – 7x + 2

Leading Coefficient The leading coefficient is the coefficient of the first term in a polynomial when the terms are written in descending order by degrees. For example, the quartic function f(x) = -2x4 + x3 – 5x2 – 10 has a leading coefficient of -2.

A polynomial function is a function of the form: Remember integers are … –2, -1, 0, 1, 2 … (no decimals or fractions) so positive integers would be 0, 1, 2 … A polynomial function is a function of the form: n must be a positive integer All of these coefficients are real numbers The degree of the polynomial is the largest power on any x term in the polynomial.

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 the graph of a polynomial This IS NOT the graph of a polynomial

Let’s look at the graph of where n is an even integer. and grows steeper on either side Notice each graph looks similar to x2 but is wider and flatter near the origin between –1 and 1 The higher the power, the flatter and steeper

Let’s look at the graph of where n is an odd integer. Notice each graph looks similar to x3 but is wider and flatter near the origin between –1 and 1 and grows steeper on either side The higher the power, the flatter and steeper

Polynomial Function in General Form Polynomial Functions Polynomial Function in General Form Degree Name of Function Constant 1 Linear 2 Quadratic 3 Cubic 4 Quartic 5 Quintic Teachers: This definition for ‘degree’ has been simplified intentionally to help students understand the concept quickly and easily. The largest exponent within the polynomial determines the degree of the polynomial.

Write the polynomials in standard form. Classify by degree and terms.  

The degree of a polynomial function affects the shape of its graph and determines the maximum number of turning points, or places the graph changes directions. It also affects the end behavior, or the directions of the graph to the far left and the far right.

POLYNOMIAL FUNCTIONS f(x) = 3 Constant Function Degree = 0 GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = 3 Constant Function Degree = 0 Max. Zeros: 0

POLYNOMIAL FUNCTIONS f(x) = x + 2 Linear Function Degree = 1 GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x + 2 Linear Function Degree = 1 Max. Zeros: 1

POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2 Quadratic Function Degree = 2 GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2 Quadratic Function Degree = 2 Max. Zeros: 2

POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2 Cubic Function Degree = 3 GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2 Cubic Function Degree = 3 Max. Zeros: 3

POLYNOMIAL FUNCTIONS f(x) = x4 + 4x3 – 2x – 1 Quartic Function GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x4 + 4x3 – 2x – 1 Quartic Function Degree = 4 Max. Zeros: 4

POLYNOMIAL FUNCTIONS f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1 Quintic GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1 Quintic Function Degree = 5 Max. Zeros: 5

POLYNOMIAL FUNCTIONS f(x) = x2 Degree: Even Leading Coefficient: + END BEHAVIOR f(x) = x2 Degree: Even Leading Coefficient: + End Behavior: Up and up

POLYNOMIAL FUNCTIONS f(x) = -x2 Degree: Even Leading Coefficient: – END BEHAVIOR f(x) = -x2 Degree: Even Leading Coefficient: – End Behavior: Down and down

POLYNOMIAL FUNCTIONS f(x) = x3 Degree: Odd Leading Coefficient: + END BEHAVIOR f(x) = x3 Degree: Odd Leading Coefficient: + End Behavior: Down and up

POLYNOMIAL FUNCTIONS f(x) = -x3 Degree: Odd Leading Coefficient: – END BEHAVIOR f(x) = -x3 Degree: Odd Leading Coefficient: – End Behavior: Up and down

End Behavior Summary Degree Even Odd Leading Coefficient Positive Up and Up Down and Up Negative Down and Down Up and Down

Assignment Homework– p. 285 #9 – 37 odds Challenge - #47-49 (must do all 3)

Cubic Polynomials Look at the two graphs and discuss the questions given below. Graph B Graph A 1. How can you check to see if both graphs are functions? 2. How many x-intercepts do graphs A & B have? 3. What is the end behaviour for each graph? 4. Which graph do you think has a positive leading coeffient? Why? 5. Which graph do you think has a negative leading coefficient? Why?

Quartic Polynomials Look at the two graphs and discuss the questions given below. Graph B Graph A 1. How can you check to see if both graphs are functions? 2. How many x-intercepts do graphs A & B have? 3. What is the end behaviour for each graph? 4. Which graph do you think has a positive leading coeffient? Why? 5. Which graph do you think has a negative leading coefficient? Why?

Factored form & Standard form Sign of Leading Coefficient Cubic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x+1)(x+4)(x-2) Standard y=x3+3x2-6x-8 -4, -1, 2 Positive As x, y and x-, y- Domain {x| x Є R} Range {y| y Є R} y=-(x+1)(x+4)(x-2) y=-x3-3x2+6x+8 Negative As x, y- and x-, y

Factored form & Standard form Sign of Leading Coefficient Cubic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x+3)2(x-1) Standard y=x3+5x2+3x-9 -3, 1 Positive As x, y and x-, y- Domain {x| x Є R} Range {y| y Є R} y=-(x+3)2(x-1) y=-x3-5x2-3x+9 Negative As x, y- and x-, y

Factored form & Standard form Sign of Leading Coefficient Cubic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x-2)3 Standard y=x3-6x2+12x-8 2 Positive As x, y and x-, y- Domain {x| x Є R} Range {y| y Є R} y=-(x-2)3 y=-x3+6x2-12x+8 Negative As x, y- and x-, y

Factored form & Standard form Sign of Leading Coefficient Quartic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x-3)(x-1)(x+1)(x+2) Standard y=x4-x3-7x2+x+6 -2,-1,1,3 Positive As x, y and x-, y Domain {x| x Є R} Range {y| y Є R, y ≥ -12.95} y=-(x-3)(x-1)(x+1)(x+2) y=-x4+x3+7x2-x-6 Negative As x, y- and x-, y- y ≤ 12.95}

Factored form & Standard form Sign of Leading Coefficient Quartic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x-4)2(x-1)(x+1) Standard y=x4-8x3+15x2+8x-16 -1,1,4 Positive As x, y and x-, y Domain {x| x Є R} Range {y| y Є R, y ≥ -16.95} y=-(x-4)2(x-1)(x+1) y=-x4+8x3-15x2-8x+16 Negative As x, y- and x-, y- y ≤ 16.95}

Factored form & Standard form Sign of Leading Coefficient Quartic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x+2)3(x-1) Standard y=x4+5x3+6x2-4x-8 -2,1 Positive As x, y and x-, y Domain {x| x Є R} Range {y| y Є R, y ≥ -8.54} y=-(x+2)3(x-1) y=-x4-5x3-6x2+4x+8 Negative As x, y- and x-, y- y ≤ 8.54}

Factored form & Standard form Sign of Leading Coefficient Quartic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x-3)4 Standard y=x4-12x3+54x2-108x+81 3 Positive As x, y and x-, y Domain {x| x Є R} Range {y| y Є R, y ≥ 0} y=-(x-3)4 y=-x4+12x3-54x2+108x-81 Negative As x, y- and x-, y- y ≤ 0}