68 Analyzing Graphs of Polynomial Functions .ppt

6.8 Analyzing Graphs of Polynomial Functions Analyzing Graphs of Polynomial Functions The relationships between zeros, factors, solutions, and x-intercepts are summarized below. Let f x a n x n + a n – 1 x n – 1 + . . . + a 1x + a 0 be a polynomial function. The following statements are equivalent. ZERO: k is a zero of the polynomial function f. FACTOR: x – k is a factor of the polynomial f x . SOLUTION: k is a solution of the polynomial equation f x 0. X-INTERCEPT: k is an x-intercept of the graph of the polynomial function f. ZEROS, FACTORS, SOLUTIONS, AND INTERCEPTS CONCEPT SUMMARY If k is a real number, then the following is also equivalent. Using x-Intercepts to Graph a Polynomial Function Graph the function f x 1 4 x + 2 x – 1 2. Plot x-intercepts. Since x + 2 and x – 1 are factors of f x , –2 and 1 are the x-intercepts of the graph of f. Plot the points –2, 0 and 1, 0 . 5 1 1 –4 3 2 0 –1 –3 –4 y x 1 2 1 2 –12 SOLUTION 1 4 Because f x has three linear factors of the form x – k and a constant factor of , it is a cubic function with a positive leading coefficient. Plot points between and beyond the x-intercepts. Determine the end behavior of the graph. Using x-intercepts to Graph a Polynomial Function Draw the graph so it passes through the points you plotted and has the appropriate end behavior. Therefore, f x – ? – ? as x f x ? ? as x and Graph the function 1 4 x + 2 x – 1 2. Remember End-Behavior Positive L.C, Even Degree both up Positive L.C, Odd Degree Right up, Left down Negative L.C., Even Degree both point down Negative L.C., Odd Degree Left up, Right down UP – UP POSITIVE DOWN – DOWN NEGATIVE EVEN DEGREES: ODD DEGREES: RISE TO THE RIGHT POSITIVE FALLS TO THE RIGHT NEGATIVE Graph the Function Plot the intercepts. Make a table and plot some points. What are the end behaviors? What kind of function is it? Graph the Function Plot your zeros. Multiply the factors together to tell you about end behavio