Polynomial Functions www.oame.on.ca/main/files/Gr12-2007/MCT4C/MCT4C-U2L3.ppt

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.

Explore Polynomials Linear Function Quadratic Function Cubic Function Quartic Function

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.

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?

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}