1. Polynomial Functions MCT4C1

2. Polynomial Functions The largest exponent within the polynomial determines the degree of the polynomial. Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic

3. Symmetry in Polynomial Functions Line symmetry must reflect across y-axis. Rotational symmetry must rotate 180 0 about origin. Polynomial Function DegreeType of Symmetry Type Of Function y=x 2 2LineEven y=x 3 3RotationalOdd y=(x+2) 2 2neither y=x 3 +13neither

4. Explore Polynomials Linear Function Quadratic Function Cubic Function Quartic Function

5. 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) = -2x 4 + x 3 – 5x 2 – 10 has a leading coefficient of -2.

6. Cubic Polynomials Look at the two graphs and discuss the questions given below. 1. How can you check to see if both graphs are functions? 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? 2. How many x-intercepts do graphs A & B have? Graph B Graph A

7. Cubic Polynomials 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=x 3 +3x 2 -6x-8 -4, -1, 2Positive As x , y  and x  - , y  -  Domain {x| x Є R} Range {y| y Є R} Factored y=-(x+1)(x+4)(x-2) Standard y=-x 3 -3x 2 +6x+8 -4, -1, 2Negative As x , y  -  and x  - , y  Domain {x| x Є R} Range {y| y Є R} The following chart shows the properties of the graphs on the left.

8. Cubic Polynomials 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=x 3 +5x 2 +3x-9 -3, 1Positive As x , y  and x  - , y  -  Domain {x| x Є R} Range {y| y Є R} Factored y=-(x+3) 2 (x-1) Standard y=-x 3 -5x 2 -3x+9 -3, 1Negative As x , y  -  and x  - , y  Domain {x| x Є R} Range {y| y Є R} The following chart shows the properties of the graphs on the left.

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

10. Quartic Polynomials Look at the two graphs and discuss the questions given below. 1. How can you check to see if both graphs are functions? 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? 2. How many x-intercepts do graphs A & B have? Graph B Graph A

11. Quartic Polynomials 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=x 4 -x 3 -7x 2 +x+6 -2,-1,1,3Positive As x , y  and x  - , y  Domain {x| x Є R} Range {y| y Є R, y ≥ -12.95} Factored y=-(x-3)(x-1)(x+1)(x+2) Standard y=-x 4 +x 3 +7x 2 -x-6 -2,-1,1,3Negative As x , y  -  and x  - , y  -  Domain {x| x Є R} Range {y| y Є R, y ≤ 12.95} The following chart shows the properties of the graphs on the left.

12. Quartic Polynomials 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=x 4 -8x 3 +15x 2 +8x-16 -1,1,4Positive As x , y  and x  - , y  Domain {x| x Є R} Range {y| y Є R, y ≥ -16.95} Factored y=-(x-4) 2 (x-1)(x+1) Standard y=-x 4 +8x 3 -15x 2 -8x+16 -1,1,4Negative As x , y  -  and x  - , y  -  Domain {x| x Є R} Range {y| y Є R, y ≤ 16.95} The following chart shows the properties of the graphs on the left.

13. Quartic Polynomials 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=x 4 +5x 3 +6x 2 -4x-8 -2,1Positive As x , y  and x  - , y  Domain {x| x Є R} Range {y| y Є R, y ≥ -8.54} Factored y=-(x+2) 3 (x-1) Standard y=-x 4 -5x 3 -6x 2 +4x+8 -2,1Negative As x , y  -  and x  - , y  -  Domain {x| x Є R} Range {y| y Є R, y ≤ 8.54} The following chart shows the properties of the graphs on the left.

14. Quartic Polynomials Equation Factored form & Standard form X- Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x-3) 4 Standard y=x 4 -12x 3 +54x 2 -108x+81 3Positive As x , y  and x  - , y  Domain {x| x Є R} Range {y| y Є R, y ≥ 0} Factored y=-(x-3) 4 Standard y=-x 4 +12x 3 -54x 2 +108x-81 3Negative As x , y  -  and x  - , y  -  Domain {x| x Є R} Range {y| y Є R, y ≤ 0} The following chart shows the properties of the graphs on the left.

15. Common Differences xy 01 11 219 379 4205 5421 6751 0 18 60 126 216 330 18 42 66 90 114 24 Since it is a 3 rd Common Difference, the function is CUBIC. The leading coefficient is Positive. The leading coefficient can be found using: 24 = a(3!) 24 = 6a 4 = a