2. Essential Question: How do we decide for the degree of the polynomial with a variable? How do we determine the end behavior of a polynomial function?

3. CCSS: F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

4. Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

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

6. x 0 Not a polynomial because of the square root since the power is NOT an integer Determine which of the following are polynomial functions. If the function is a polynomial, state its degree. A polynomial of degree 4. A polynomial of degree 0. We can write in an x 0 since this = 1. Not a polynomial because of the x in the denominator since the power is negative

7. Graphs of polynomials are smooth and continuous. No sharp corners or cusps No gaps or holes, can be drawn without lifting pencil from paper This IS the graph of a polynomial This IS NOT the graph of a polynomial

8. 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 x 2 but is wider and flatter near the origin between –1 and 1 The higher the power, the flatter and steeper

9. Let’s look at the graph of where n is an odd integer. and grows steeper on either side Notice each graph looks similar to x 3 but is wider and flatter near the origin between –1 and 1 The higher the power, the flatter and steeper

10. Let’s graph Looks like x 2 but wider near origin and steeper after 1 and -1 Reflects about the x-axis Translates up 2 So as long as the function is a transformation of x n, we can graph it, but what if it’s not? We’ll learn some techniques to help us determine what the graph looks like in the next slides.

11. LEFT RIGHT and HAND BEHAVIOUR OF A GRAPH The degree of the polynomial along with the sign of the coefficient of the term with the highest power will tell us about the left and right hand behaviour of a graph.

12. Even degree polynomials rise on both the left and right hand sides of the graph (like x 2 ) if the coefficient is positive. The additional terms may cause the graph to have some turns near the center but will always have the same left and right hand behaviour determined by the highest powered term. left hand behaviour: rises right hand behaviour: rises

13. Even degree polynomials fall on both the left and right hand sides of the graph (like - x 2 ) if the coefficient is negative. left hand behaviour: falls right hand behaviour: falls turning points in the middle

14. Odd degree polynomials fall on the left and rise on the right hand sides of the graph (like x 3 ) if the coefficient is positive. left hand behaviour: falls right hand behaviour: rises turning Points in the middle

15. Odd degree polynomials rise on the left and fall on the right hand sides of the graph (like x 3 ) if the coefficient is negative. left hand behaviour: rises right hand behaviour: falls turning points in the middle

16. A polynomial of degree n can have at most n-1 turning points (so whatever the degree is, subtract 1 to get the most times the graph could turn). doesn’t mean it has that many turning points but that’s the most it can have Let’s determine left and right hand behaviour for the graph of the function: degree is 4 which is even and the coefficient is positive so the graph will look like x 2 looks off to the left and off to the right. The graph can have at most 3 turning points How do we determine what it looks like near the middle?

17. x and y intercepts would be useful and we know how to find those. To find the y intercept we put 0 in for x. (0,30) To find the x intercept we put 0 in for y. Finally we need a smooth curve through the intercepts that has the correct left and right hand behavior. To pass through these points, it will have 3 turns (one less than the degree so that’s okay)

18. We found the x intercept by putting 0 in for f(x) or y (they are the same thing remember). So we call the x intercepts the zeros of the polynomial since it is where it = 0. These are also called the roots of the polynomial. Can you find the zeros of the polynomial? There are repeated factors. (x-1) is to the 3 rd power so it is repeated 3 times. If we set this equal to zero and solve we get 1. We then say that 1 is a zero of multiplicity 3 (since it showed up as a factor 3 times). What are the other zeros and their multiplicities? -2 is a zero of multiplicity 2 3 is a zero of multiplicity 1

19. So knowing the zeros of a polynomial we can plot them on the graph. If we know the multiplicity of the zero, it tells us whether the graph crosses the x axis at this point (odd multiplicities CROSS) or whether it just touches the axis and turns and heads back the other way (even multiplicities TOUCH). Let’s try to graph: You don’t need to multiply this out but figure out what the highest power on an x would be if multiplied out. In this case it would be an x 3. Notice the negative out in front. What would the left and right hand behavior be? What would the y intercept be? (0, 4) Find the zeros and their multiplicity 1 of mult. 1 (so crosses axis at 1) -2 of mult. 2 (so touches at 2)

20. Steps for Graphing a Polynomial Determine left and right hand behaviour by looking at the highest power on x and the sign of that term. Determine maximum number of turning points in graph by subtracting 1 from the degree. Find and plot y intercept by putting 0 in for x Find the zeros (x intercepts) by setting polynomial = 0 and solving. Determine multiplicity of zeros. Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behavior as a guide.

21. Let’s graph: Determine left and right hand behavior by looking at the highest power on x and the sign of that term. Multiplying out, highest power would be x 4 Determine maximum number of turns in graph by subtracting 1 from the degree. Degree is 4 so maximum number of turns is 3 Find and plot y intercept by putting 0 in for x Find the zeros (x intercepts) by setting polynomial = 0 and solving. Zeros are: 0, 3, -4 Determine multiplicity of zeros.0 multiplicity 2 (touches) 3 multiplicity 1 (crosses) -4 multiplicity 1 (crosses) Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behaviour as a guide. Here is the actual graph. We did pretty good. If we’d wanted to be more accurate on how low to go before turning we could have plugged in an x value somewhere between the zeros and found the y value. We are not going to be picky about this though since there is a great method in calculus for finding these maxima and minima.

22. What is we thought backwards? Given the zeros and the degree can you come up with a polynomial? Find a polynomial of degree 3 that has zeros –1, 2 and 3. What would the function look like in factored form to have the zeros given above? Multiply this out to get the polynomial. FOIL two of them and then multiply by the third one.

23. 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

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

25. 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 behavior for each graph? 4. Which graph do you think has a positive leading coefficient? 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

26. 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.

27. 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.

28. 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.

29. 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 behavior for each graph? 4. Which graph do you think has a positive leading coefficient? 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

30. 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.

31. 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.

32. 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.

33. 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.