3.5 Graphs of Polynomial Functions

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1 . Graphs of olynomial Functions Symmetry of olynomial Functions: This information is a review of symmetry from the unit on graphs of functions. We W will be considering two types of symmetry in this lesson; symmetry about the y- ais and symmetry about the origin. Even Functions: An even function is a function that i functions with all even eponents ar f g 1 h Although it is clear by inspection that the above polynomial functions are even, such s is not always the case. To determine if a function is even we may substitute the -values with -values and determine if f f. Eample: Show that the function f Solution: Replace all s with s.. f f f f f Since f f the function is even and is therefore symmetric about the y-ais. is symmetric to the y-ais. It gets its name from the fact that polynomial re symmetric to the y-ais. Some eamples of o even functions are: is even. Eample: Show that the function h is even. Solution: Replace all s with s. h h h h h Since h h the function is even and is therefore symmetric about the y-ais.

2 Odd Functions: An odd function is a function that iss symmetric to the origin. It gets its name fromm the fact that polynomial functions with all odd eponents aree symmetric to the origin. This type of symmetry is also called a 10 degree rotation since all points on the graph are rotated 10 degrees about the origin. Some eamples of odd functions are: f g 1 h Although it is clear by inspection that the above polynomial functions are odd, such is not always the case. To determine if a function is odd wee may substitute the -values with -values and determine if f f. Eample: Show that the function h is odd. Solution: Replace all s with s.. h h h h h h Since h h the function is odd and is therefore symmetric about the origin. Eample: Show that the function g 1 is odd. Solution: Replace all s with s. g 1 g g g 1 1 g g Since g g the function is odd and is therefore symmetric about the origin. 1

3 Behavior at the X-Intercepts: When a polynomial equation touches the -ais, it will do one of two things; cross the -ais or not cross the -ais. This behavior is determined from the factors of the polynomial as follows: 1. The graph crosses the -ais is the factor the root comes from is raised to an odd power odd multiplicity.. The graph does not cross the -ais turns around if the factor the root comes from is raised to an even power even multiplicity. Eample: Determine the behavior of the polynomial 1 at its roots. Solution: Since this polynomial is already factored, we simply derive the roots from the factors. Factor Root K Behavior Crosses Not Cross Remember that K represents the multiplicity of the factor or how many times it occurs. Eample: Determine the behavior of the polynomial at its roots. Solution: Since this polynomial is already factored, we simply derive the roots from the factors. Factor Root K - - Behavior Crosses Not Cross Remember that K represents the multiplicity of the factor or how many times it occurs.

4 Eample: Determine the behavior of the polynomial 10 at its roots. Solution: After using synthetic division to determine the factors, the polynomiall may be written as 1. Now we simply derive the roots from the factors. Factor Root K Behavior Crosses Not Cross Remember that K represents the multiplicity of the factor or how many times it occurs. Turning oints: A polynomial function of degree n will have at most n-1 turning points. Eample: The function f points. 1is a rd degree polynomial, so it will have at most turning Eample: The function f 1 is a th degree polynomial, so it will have at most turning points.

5 End Behavior of a olynomial Function: A polynomial function has very predictable behavior as the -values approach positive or negative infinity and is refered to as the functions end behavior. To state that approaches infinity means that is increasing without bound. Likewise, to state that approaches negative infinity means that is decreasing without bound. As, the y-values of the polynomial function will increase or decrease without bound. Likewise, as, the y-values of the polynomial function will increase or decrease without bound. This behavior can be predicted by looking at the leading term of the polynomial. Specifically we need to look at the sign and the power of the leading term. This is called the Leading Term Test: Eample: Determine the end behavior of the polynomial function 1 Solution: We only need to look at the leading term. Since the term is positive and the power is odd, we have:

6 Eample: Determine the end behavior of the polynomial function Solution: We only need to look at the leading term since the term is negative and the power is odd, we have: Eample: Determine the end behavior of the polynomial function Solution: We only need to look at the leading term. Since the term is positive and the power is even, we have: Eample: Determine the end behavior of the polynomial function Solution: We only need to look at the leading term. Since the term is negative and the power is even, we have:. Sketching the Graph of a olynomial

7 Function: To sketch the graph of a polynomial we will do the following: 1. Determine number of roots and turning points from leading term.. Find the roots of the equation and their multiplicity. Determine the end behavior of the function. Determine the symmetry, if any.. Graph the equation. Eample: Sketch the graph of the polynomial function Solution: Step 1: Find the zeros of the polynomial equation. Using synthetic division the equation may be written in terms of its factors as: Step : Determine nature of function at its roots Factor Root K Behavior Crosses - Not Cross Step : Determine nature of function as have: and. Since the leading term is positive and odd we

8 The Derivative: A useful mathematical tool borrowed from calculus is the derivative. To find the derivative of a polynomial function, multiply the coefficient of each term by the eponent and reduce the value of the eponent by one. If the original function is then the new function is pronounced prime of.. Eample: Find the derivative of the function Solution: Multiply the coefficient of each term by the eponent and reduce the value of the eponent by one. 10 ' Eample: Find the derivative of the function 1 Solution: Multiply the coefficient of each term by the eponent and reduce the value of the eponent by one ' 1 Eample: Find the derivative of the function 1 Solution: Multiply the coefficient of each term by the eponent and reduce the value of the eponent by one. 1 1 ' 1 Eample: Find the derivative of the function Solution: Multiply the coefficient of each term by the eponent and reduce the value of the eponent by one. 9 1 ' Eample: Find the derivative of the function Solution: Multiply the coefficient of each term by the eponent and reduce the value of the eponent by one. '

9 Local Maimum and Minimum Values: A useful method for finding the local maimum and minimum values of a higher degree e polynomial function is to use the derivative. This method has three steps: 1. Find the derivative.. Set the derivative equal to zero and solve the equation.. Evaluate the -values in the find the y-values. original function to Eample: Find the local maimum and minimum values of the function 9 0 Solution: Find the derivative. ' Set the derivative equal to zero and solve the quadratic equation ' Evaluate the -values in the original function to find the y-values Therefore, the local minimum is, - and the local maimum is -,.

10 Finding Function from Graph: The graph of a polynomial function can be used to find a function that describes the graph. Eample: Given the graph of a polynomial function, determine a function which could describe it. Be sure to identify each of the following: 1. End Behavior. Roots & Multiplicity. # of Turning oints and n.. Factors of the olynomial. Equation of olynomial Solution: 1. End Behavior: Starts low, ends high so according to the leading term test it has a positive, odd leading term.. Roots, Multiplicity & Turning oints:, k 1 0, k, k. Turning oints: There are turning points, therefore, n.. Using the roots, determine the factors and multiply to find the function Therefore, a function is f 0

11 .-Applications Eample: A company s weekly profit in thousands of dollars is given by the function Where is the amount in thousands of dollars spent per week on advertising. Determine the local maimum and minimum value for the profit. Solution: To determine the local maimum and minimum values for profit, I will use the derivative. 0 ' Solve the quadratic equation ± ± To find the y-values which represent revenue, evaluate these two points in the original polynomial function The local maimum profit is $,.00 when $0.00 is spent on advertising. The local minimum is $,1.00 when $1,0.00 is spent on advertising.

12 Eample: An open-top bo is to be made from a in. by in. piece of copper by cutting equal squares from each corner and folding up the sides. Write the volume of the bo as a function of. Determine the maimum possible volume of the bo. Solution: The formula for the volume of a rectangular bo is VLWH. Let length of square cut from the bo. We can then create the following equation. V V V ' 1 Using the quadratic equation we obtain 1.0 and.. However, since the width of the copper sheet is only in. it would be impossible to cut two. in sections out of the corners. Therefore the only legitimate solution is 1.0. The actual volume is: V V The volume of the bo is 0.1 in.cu.

13 Eample: Find a function that represents the given graph: Solution: 1. End Behavior: Starts low, ends high so according to the leading term test it has a positive, odd leading term.. There are turning points; therefore, the function will be a th degree polynomial. A polynomial function always has one lesss turning point than the degree of the function.. Roots & Multiplicity:, k 0, k 1, k 1. Using the roots and the value of a, determine the factors and multiply to find the function. f 1 f f f Therefore, the function is f 9