UNIT 8 STUDY SHEET POLYNOMIAL FUNCTIONS

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1 UNIT 8 STUDY SHEET POLYNOMIAL FUNCTIONS KEY FEATURES OF POLYNOMIALS Intercepts of a function o x-intercepts - a point on the graph where y is zero {Also called the zeros of the function.} o y-intercepts - a point on the graph where x is zero Relative max/min o Relative maximum - The point(s) on the graph, which have maximum y- values or second coordinates relative to the points close to them on the graph. o Relative minimum - The point(s) on the graph, which have minimum y-values or second coordinates relative to the points close to them on the graph. o These relative extrema (turning points) help determine the intervals for which a function is increasing or decreasing Increasing/Decreasing Functions o A function is said to be increasing when the y-value increases as the x-value increases. o A function is said to be decreasing when the y-value decreases as the x-value increases. o The graph must be read from left-to-right to determine if it is increasing or decreasing. Increasing: Decreasing:

2 DEGREE OF A POLYNOMIAL AND END BEHAVIOR End Behavior (Let f be a function whose domain and range are subsets of the real numbers. The end behavior of a function f is a description of what happens to the y-values of the function o as approaches positive infinity, and o as approaches negative infinity.

3 Sketching Graphs of Polynomials in Factored Form By looking at the factored form of a polynomial, we can identify important characteristics of the graph such as -intercepts and degree of the function, which in turn allow us to develop a sketch of the graph. A polynomial function of degree may have up to -intercepts. A polynomial function of degree may have up to relative maxima and minima. How to Graph Polynomials in Factored Form 1. Zeros Factor the polynomial to find all its real zeros; these are the x-intercepts of the graph. 2. Test Points Test a point between the x-intercepts to determine whether the graph of the polynomial lies above or below the x-axis on the intervals determined by the zeros. 3. End Behavior Determine the end behavior of the polynomial by looking at the degree of the polynomial and the sign of the leading coefficient. 4. Graph Plot the intercepts and other points you found when testing. Sketch a smooth curve that passes through these points and exhibits the required end behavior.

4 EVEN FUNCTIONS ODD AND EVEN FUNCTIONS ODD FUNCTIONS An EVEN function has the following properties: I. Its graph is symmetric about the y-axis (reflection over the y-axis) II. f( ) f( ) Trick: Can you fold the graph in half along the y-axis and it aligns perfectly? Then it is even. An ODD function has the following properties: I. Its graph is symmetric about the origin (reflection over the point (0, 0) II. f( ) f( ) Trick: Can you rotate the graph upside down and it still looks the same? Then it is odd. To determine if a function is odd or even when looking at the graph, look for symmetery! If symmetric about the y-axis, it is even If symmetric about the origin, it is odd

5 ODD AND EVEN FUNCTIONS (Continued) To determine if a function is odd or even algebraically, substitute given function! If ( ) ( ), it is even If ( ) ( ), it is odd into the To determine algebraically if a fraction is an odd or even function, you must factor out a -1 in the numerator and denominator (if possible). For example:

6 Synthetic Division, Remainder Theorem, Factor Theorem:

7 Using Synthetic Division to Find all Factors and Zeros Steps: 1. Determine a factor of the polynomial (from the graph or given in the question) and prove algebraically it is a factor. 2. Use synthetic division to find a quotient that is also a factor of the original polynomial. 3. Use this new polynomial (quotient) to find the remaining zeros by using an appropriate method! 1. ( ) ; x +3 is a factor of ( ) 2. ( ) ; ( )

8 Zeros and Multiplicity of Polynomials The real zeros of a polynomial function may be found by factoring (where possible) or by finding where the graph touches the x-axis. The number of times a zero occurs is called its multiplicity. In an equation, the multiplicity of the zero is represented by the exponent of each factor. The multiplicity of a zero tells you how the graph will interact with the x-axis at a specific zero. Odd Multiplicity The graph crosses the x-axis at the given zero. Even Multiplicity The graph is tangent to the x-axis at the given zero. Example: Below is the graph of ( ) ( ). Example: Below is the graph of ( ) ( ). You can see the graph crosses the x-axis at the zero x = 3. You can see the graph is tangent to the x-axis at the zero x = 3. Examples: Find the zeros of the following polynomial functions, with their multiplicities. a. ( ) ( )( )( ) with multiplicity with multiplicity b. ( ) ( ) ( ) with multiplicity with multiplicity c. ( ) ( ) with multiplicity

9 USING ZEROS TO WRITE POLYNOMIAL EQUATIONS We will use the following steps to write a polynomial function from its given zeros: 1. Convert the zeros to factors. 2. Multiply the factors. 3. Combine like terms and write with powers of x in descending order, which is the standard form of a polynomial function. REMEMBER: If one zero is irrational or imaginary, another zero must be its conjugate! For Example: If a given zero is, then another zero must be For Example: If a given zero is, then another zero must be

10 WRITING POLYNOMIAL EQUATIONS TO MODEL A GIVEN SITUATION Steps: Find the x-intercepts Write the equation in factored form using the x-intercepts and a constant factor c. Plug a point from the graph into the equation to find the value of c. Write equation with c plugged back in to the factored form.