Graphs of Polynomials: Polynomial functions of degree 2 or higher are smooth and continuous. (No sharp corners or breaks).

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1 Graphs of Polynomials: Polynomial functions of degree or higher are smooth and continuous. (No sharp corners or breaks). These are graphs of polynomials. These are NOT graphs of polynomials There is a break and a hole. There is a corner and a sharp turn. Name and Shape of Polynomials: Polynomials are named by their degree. The degree is the largest eponent, after it has been completely distributed. Degree of Polynomial Name Shape of Graph 0 Constant Line 1 Linear Line Quadratic Parabola Cubic Smooth Curve Quartic Smooth Curve 5 Quintic Smooth Curve 6 or more nth Degree (n is the degree) Smooth Curve Polynomials are also named by the number of terms after the polynomial has been completely distributed. Degree of Polynomial Number of Terms 1 Monomial Binomial Trinomial or more Polynomial A polynomial function is in standard form if it is completely distributed and the terms are arranged in descending order by degree. Eample) Put the polynomial in standard form. Classify it by degree and number of terms. a) f ( ) 5 Standard form: 5 Name: Quartic Trinomial (degree ; terms) 5 b) 10 5 Standard form: 1 Name: Quintic Trinomial (degree 5; terms)

2 End Behavior Eploration Activity: Graph each function on the calculator. Determine the end behavior of f() as approaches negative and positive infinity. Fill in the table and write your conclusion regarding the degree of the function and the end behavior. f () Degree Sign of Leading End Behavior in words End Behavior in Limit Notation Coefficient UP-UP DOWN-DOWN As DOWN-UP As UP-DOWN As UP-UP As DOWN-DOWN As 5 5 DOWN-UP As 5 5 UP-DOWN As 6 6 UP-UP As 6 6 DOWN-DOWN As

3 End Behavior Conclusion: This describes what the function looks like at the far left and at the far right of the graph. The leading coefficient is the number in front of the variable with the largest eponent. It tells us about the end behavior. We use limit notation to describe end behavior. We describe the far left side of the graph as and the far right side of the graph as. We describe the graph going up as f () and going down as f (). In words Limit Notation UP left UP right DOWN left UP right DOWN left DOWN right UP left DOWN right Eample) Determine the end behavior. Write your answer in limit notation. a) 5 c) ( 5)( 1) ( ) 1 b) d) ( 5)( 1) ( )

4 Zeros (-intercepts) and their multiplicity: Zeros are -intercepts. They tell us where the graph touches the -ais. Find -intercepts by setting the equation equal to zero and solving for. Multiplicity is the number of times a number is a zero. So if ( h) k 0, then h is a zero of multiplicity k. If a zero has an ODD multiplicity, then the graph CROSSES the -ais at the zero. If the zero has an EVEN multiplicity, then the graph will BOUNCE off the -ais (approach it, touch it, and turn away from it). Turning Points: A turning point is a point on the graph where the graph changes from increasing to decreasing or from decreasing to increasing. The y-coordinate of a turning point is called a local or relative maimum if the point is higher than all nearby points. The y-coordinate of a turning point is called a local or relative minimum if the point is lower than all nearby points. Global or absolute maimums or minimums are the greatest or least values of the entire graph. A graph has AT MOST n-1 turning points. If a polynomial of degree n has n distinct real zeros, then the graph has EXACTLY n-1 turning points. Eample: Determine the sign of the leading coefficient and the least possible degree of the polynomial. a) Leading coefficient negative (right side down) Degree is at least (it has 0 turning points so the degree is at least 1. It s not a line, so it can t be 1. It s odd, so it can t be.) b) Leading coefficient positive (right side up) Degree is at least (it has turning points and is even)

5 Zeros and Turning Points Eploration: Use a graphing calculator to graph the cubic functions. Then use the graph of each function to answer the questions in the table. Function ( ) ( )( ) How many distinct factors 1 does f () have? What are the graph s - 0 0, 0,,- intercepts? Is the graph tangent to the -ais or does it cross the - ais at each -intercept? Crosses at =0 Tangent at =0; Crosses at = Crosses at =0; Crosses at =; Crosses at =- How many turning points 0 does the graph have? How many global maimum values? How many local maimum values that are not global? How many global minimum values? How many local minimum values that are not global? No maimum values No minimum values No global maimum values; 1 local maimum value No global minimum values; 1 local minimum value No global maimum values; 1 local maimum value No global minimum values; 1 local minimum value Function ( ) ( )( ) ( )( )( ) How many 1 distinct factors? What are the - 0 0, 0,,- 0,,-,- intercepts? Is the graph tangent to the - ais or does it cross the -ais Tangent at =0 Crosses at =0; Crosses at = Tangent at =0; Crosses at =; Crosses at =- Crosses at =0; Crosses at =; Crosses at =-; Crosses at =- at each - intercept? How many 1 1 turning points? How many global maimum values? How many local maimum values that are not global? How many global minimum values? How many local minimum values that are not global? No maimum values 1 global minimum value; No local minimum values No maimum values 1 global minimum value; No local minimum values No global maimum values; 1 local maimum value 1 global minimum value (which occurs twice); No local minimum values No global maimum values; 1 local maimum value 1 global minimum value; 1 local minimum values

6 Sketching the graph of a polynomial: Given a polynomial in intercept form (fully factored), you can sketch the graph using the end behavior, zeros, and the sign of the function values on intervals determined by the -intercepts. The sign of the function values tells you whether the graph is above or below the -ais on a particular interval. You can determine the sign of the function by determining the sign of each factor and recognizing what the sign of the product of those factors is. Eample) Sketch the graph of the polynomial ( )( ). Find end behavior: The polynomial has degree. The leading coefficient is positive. So the end behavior is DOWN-UP. Written in limit notation, we say as, f() and as,f(). Find the number of turning points: The polynomial has degree. There are distinct zeros. So there are eactly turning points. Find the zeros and multiplicity: The zeros are =0 (multiplicity 1), =- (multiplicity 1), = (multiplicity 1). Find the y-intercept: The y-intercept is the y value when =0. So, (0)(0+)(-)=0. The y-intercept is (0,0). Find the sign of the function on the intervals defined by its zeros. Interval Sign of the Sign of Sign of (+) Sign of (-) Sign of f() Constant Factor (, ) (,0) ( 0,) (, ) With a sketch, you should be precise about where the graph crosses the -ais and the y-ais, but you do not need to be precise about the y-coordinates of the turning points.