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5.4 - Analyzing Graphs of Polynomial Functions Day 1 Example: f(x) = –x3 – 4x2 + 5 by making a table of values. Graph Example: Which graph is the graph of f(x) = x3 + 2x2 + 1? A. B. C. D. Location Principle: Example: Determine consecutive values of x between which each real zero of the function f(x) = x4 – x3 – 4x2 + 1 is located. Then draw the graph. Maximum & Minimum Points Relative Maximum – a point on the graph of a function where no other nearby points have a greater y-coordinate. Relative Minimum - a point on the graph of a function where no other nearby points have a lesser y-coordinate. Maximum & Minimum Points Extrema Turning – max. and min. values of a function. Point – when the graph turns. Another name for relative max. and min. - The graph of a polynomial function of degree n has at most n – 1 turning points. Example: f(x) = x3 – 3x2 + 5. Find the x-coordinates at which the relative maxima and relative minima occur. Graph Find Extrema on Calculator: Enter equation into y =. 2nd Calc Choose 3: minimum or 4: maximum. Curser on left of min/max, enter. Curser on right of min/max, enter. Enter. Example: the graph of f(x) = x3 + 3x2 + 2. Estimate the x-coordinates at which the relative maximum and relative minimum occur. Consider A. relative minimum: x = 0 relative maximum: x = –2 B. relative minimum: x = –2 relative maximum: x = 0 C. relative minimum: x = –3 relative maximum: x = 1 D. relative minimum: x = 0 relative maximum: x = 2 5.4 - Analyzing Graphs of Polynomial Functions Day 2: Real-World Problems Example: a. The weight w, in pounds, of a patient during a 7-week illness is modeled by the function w(n) = 0.1n3 – 0.6n2 + 110, where n is the number of weeks since the patient became ill. Graph the equation. Example: b. Describe the turning points of the graph and its end behavior. c. What trends in the patient’s weight does the graph suggest? d. Is it reasonable to assume the trend will continue indefinitely? Example: The graph models the cross section of Mount Rushmore. What is the smallest degree possible for the equation that corresponds with this graph? Example: The rainfall r, in inches per month, in a Midwestern town during a 7-month period is modeled by the function r(m) = 0.01m3 – 0.18m2 + 0.67m + 3.23, where m is the number of months after March 1. Graph the equation. A. B. C. D. Example: Describe the turning points of the graph and its end behavior. A. There is a relative minimum at Month 2. r(m) decreases as m increases. B. There is a relative maximum at Month 2. r(m) decreases as m increases. C. There is a relative maximum at Month 2. r(m) increases as m increases. D. There is a relative minimum at Month 2. r(m) decreases as m decreases. Example: What trends in the amount of rainfall received by the town does the graph suggest? A. The rainfall decreased the first two months, then increased. B. The rainfall increased the first two months, then decreased. C. The rainfall continued to increase throughout the entire 8 months. D. The rainfall continued to decrease throughout the entire 8 months. Example: Is it reasonable to assume the trend will continue indefinitely? A. yes B. no