Exam 2 Review

1. Complex Numbers (graphing and symbolic manipulation) • Using the graph of a function find the completely factored form. • Using a polynomial in completely factored form, graph the function. • Understand and use x intercepts (zeros) and their multiplicities. • Understand and use end behavior • Symbolically determine if a function is odd, even, or neither odd nor • even. • Symbolically and graphically find vertical asymptotes, horizontal • asymptotes and holes • Use vertical asymptotes, horizontal asymptotes, holes and degree to • create graphs. • Find local, and absolute maximums and minimums • Given a graph, find the minimum possible degree of a polynomial • using the number of turns. • Solve radical equations. • Perform Cubic and/or Quartic regression on TI calculator Exam 2 Review

2. Complex Numbers (graphing and symbolic manipulation) Addition of complex numbers Multiplication of complex numbers

3. For the function The degree is 5The x-intercepts(zeros) are x=-2,4The y-intercept is (0,8)Graph the function. Odd function with positive lead coefficient means f(x) goes to minus infinity as x goes to negative infinity and f(x) goes to plus infinity as x goes to positive infinity. y-intercept at (0,8) x-intercept at -2 is an cubed multiplicity x-intercept at 4 is a squared multiplicity

4. Given the following graph: Find the completely factored form

5. Given the following graph: Is the degree of this function odd or even? Is the leading coefficient positive or negative? What is the minimum possible degree of this function? # turns +1 = 3

6. Graph the following function on your calculator. How many real zeros does this function have? 2 How many imaginary zeros does this function have? 4-2=2

7. Solve these equations:

8. Finding Extrema Graphs of polynomial functions often have “hills” or “valleys”. The “highest hill” on the graph is located at (–2, 12.7). This is the absolute maximum of g. There is a smaller peak located at the point (3, 2.25). This is called the local maximum. Maximum and minimum values are called extrema.

9. Finding Extrema A function may have several local extrema, but only one absolute maximum and one absolute minimum. It is possible for a function to assume an absolute extremum at two values of x. The absolute maximum is 11. 11 is a local maximum as well, because nearx = –2 and x = 2 it is the largest y value.

10. Graph using calculator and find the following four features: Hole: Vertical Asymptotes: Horizontal Asymptote: X-intercepts : x=-3 x=5 and x=-2 y=0, Degree of denominator is more than the degree of the numerator x=4

11. Even Symmetry If a graph was folded along the y-axis, and the right and left sides match, then the graph would be symmetric with respect to the y-axis. A function whose graph satisfies this characteristic is called an even function. A function f is an even function if f(–x) = f(x) for every x in its domain. The graph of an even function is symmetric with respect to the y-axis. To test for even start with

12. Example: Neither Odd nor Even A function f is an even function if f(–x) = f(x) for every x in its domain. The graph of an even function is symmetric with respect to the y-axis. A function f is an odd function if f(–x) = –f(x) for every x in its domain. The graph of an odd function is symmetric with respect to the origin. To test for even or odd start with This is not f(x) or -f(x) So it is neither odd nor even !!!