Polynomial Functions Day 1 and 2

Polynomial Functions Do now: Find the Range of yesterday’s exit ticket problem! Objectives: Given a polynomial (many number) function, determine from the graph what degree is Find the “zeros” from the graph or the equation, in order to recognize equations of same degree Exit Ticket: Start homework: Do not lose handout-hmwk #5

Polynomial Functions (day 2) Do now: Write the TWO fundamental Rules of Algebra you (memorized?!) learned from yesterday’s powerpoint….! Objectives: Given a polynomial (many number) function, determine from the graph what degree is Find the “zeros” from the graph or the equation, in order to recognize equations of same degree Exit Ticket: Sketch an exponential GROWTH function

Just a few definitions  Polynomial function a function with one or more terms Ex) 2x5 – 5x3 – 10x + 9, because it has 4 terms. Ex) 7x4 + 6x2 + x has 3 terms Degree the highest exponent power that is in a term Ex) 5 x3 has a degree of “3” Ex) 10x6 has a degree of “6” Highest Degree Highest degree that is in a polynomial. When you are asked for the degree of a polynomial, you are being asked for the highest degree. Ex) 2x5 – 5x3 – 10x + 9 has a highest degree of 5 Ex) 7x4 + 6x2 + x has a highest degree of 4

Polynomial Function in General Form Polynomial Functions Polynomial Function in General Form Degree Name of Function 1 Linear 2 Quadratic 3 Cubic 4 Quartic 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.

Fundamental Theorem of Algebra: Degree of the polynomial is the same as the number of “ups” and “downs” of its graph… Try the examples in notes.

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 cubic function f(x) = -2x3 + x2 – 5x – 10 has a leading coefficient of -2. This will play an important role in it’s graph…

2nd Fundamental Theorem of Algebra: The number of zeros that a polynomial function has is equal to that function’s degree.

Explore Polynomials Linear Function Quadratic Function Cubic Function Quartic Function

Cubic Polynomials Let’s look at the two graphs and let’s discuss the questions 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

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 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}

Polynomial Functions Did we accomplish our objectives? Objectives: Given a polynomial (many number) function, determine from the graph what degree is Find the “zeros” from the graph or the equation, in order to recognize equations of same degree Any Questions?