Graphs of Polynomial Functions Section 5 3 Polynomial

Graphs of Polynomial Functions Section 5 -3

Polynomial Functions Polynomial Function in General Form Degree Name of Function 1 2 3 4 Linear Quadratic Cubic Quartic 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) = -2 x 4 + x 3 – 5 x 2 – 10 has a leading coefficient of -2.

Cubic Polynomials Look at the two graphs and discuss the questions given below. Graph A Graph B 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?

Increasing, Decreasing, Constant Intervals A function f is increasing on an interval if as x increases, then f(x) increases. A function f is decreasing on an interval if as x increases, then f(x) decreases. A function f is constant on an interval if as x increases, then f(x) remains the same.

End Behavior of Functions The end behavior of a graph describes the far left and the far right portions of the graph. Using the leading coefficient and the degree of the polynomial, we can determine the end behaviors of the graph. This is often called the Leading Coefficient Test.

End Behavior of Functions First determine whether the degree of the polynomial is even or odd. degree = 2 so it is even Next determine whether the leading coefficient is positive or negative. Leading coefficient = 2 so it is positive

END BEHAVIOR Degree: Even Leading Coefficient: End Behavior: Up Up +

END BEHAVIOR Degree: Even Leading Coefficient: End Behavior: Down

END BEHAVIOR Degree: Odd Leading Coefficient: + End Behavior: Down Up

END BEHAVIOR Degree: Odd Leading Coefficient: End Behavior: Up Down

END BEHAVIOR PRACTICE Give the End Behavior:

END BEHAVIOR PRACTICE Give the End Behavior: Up Down

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 Factored y=(x+1)(x+4)(x-2) Standard -4, -1, 2 Positive As x , y and x - , y - Negative As x , y - and x - , y y=x 3+3 x 2 -6 x-8 Factored y=-(x+1)(x+4)(x-2) Standard y=-x 3 -3 x 2+6 x+8 -4, -1, 2 End Behaviour Domain and Range Domain {x| x Є R} Range {y| y Є R}

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 Factored y=(x+3)2(x-1) Standard -3, 1 Positive y=x 3+5 x 2+3 x-9 Factored y=-(x+3)2(x-1) Standard y=-x 3 -5 x 2 -3 x+9 -3, 1 Negative End Behaviour As x , y and x - , y - As x , y - and x - , y Domain and Range Domain {x| x Є R} Range {y| y Є R}

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 Factored y=(x-2)3 Standard 2 Positive y=x 3 -6 x 2+12 x-8 Factored y=-(x-2)3 Standard y=-x 3+6 x 2 -12 x+8 2 Negative End Behaviour As x , y and x - , y - As x , y - and x - , y Domain and Range Domain {x| x Є R} Range {y| y Є R}

Quartic Polynomials Look at the two graphs and discuss the questions given below. Graph A Graph B 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 The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form XIntercepts Sign of Leading Coefficient Factored y=(x-3)(x-1)(x+2) Standard -2, -1, 1, 3 Positive y=x 4 -x 3 -7 x 2+x+6 Factored y=-(x-3)(x-1)(x+2) Standard y=-x 4+x 3+7 x 2 -x-6 -2, -1, 1, 3 Negative End Behaviour As x , y and x - , y As x , y - and x - , y - Domain and Range Domain {x| x Є R} Range {y| y Є R, y ≥ -12. 95} Domain {x| x Є R} Range {y| y Є R, y ≤ 12. 95}

Quartic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form XIntercepts Sign of Leading Coefficient Factored y=(x-4)2(x-1)(x+1) Standard -1, 1, 4 Positive y=x 4 -8 x 3+15 x 2+8 x-16 Factored y=-(x-4)2(x-1)(x+1) Standard y=-x 4+8 x 3 -15 x 2 -8 x+16 -1, 1, 4 Negative End Behaviour As x , y and x - , y As x , y - and x - , y - Domain and Range Domain {x| x Є R} Range {y| y Є R, y ≥ -16. 95} Domain {x| x Є R} Range {y| y Є R, y ≤ 16. 95}

Quartic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form XIntercepts Sign of Leading Coefficient Factored y=(x+2)3(x-1) Standard -2, 1 Positive y=x 4+5 x 3+6 x 2 -4 x-8 Factored y=-(x+2)3(x-1) Standard y=-x 4 -5 x 3 -6 x 2+4 x+8 -2, 1 Negative End Behaviour As x , y and x - , y As x , y - and x - , y - Domain and Range Domain {x| x Є R} Range {y| y Є R, y ≥ -8. 54} Domain {x| x Є R} Range {y| y Є R, y ≤ 8. 54}

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