Warm Up What do you know about the graph of f(x) = (x – 2)(x – 4)2 ?

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Presentation transcript:

Warm Up What do you know about the graph of f(x) = (x – 2)(x – 4)2 ?

5.4 - Analyzing Graphs of Polynomial Functions Day 1

Location Principle:

Example: Determine consecutive values of x between which each real zero of the function f(x) = x4 – x3 – 4x2 + 1 is located.

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 – max. and min. values of a function. Turning Point rules to remember The graph of a polynomial function of degree n has at most n – 1 turning points. If 2+ turning points are between roots, imaginary numbers exist

Example: Graph f(x) = x3 – 3x2 + 5. Find the x-coordinates at which the relative maxima and relative minima occur.

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.

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 by creating a table of values. Describe the turning points of the graph and its end behavior. Is it reasonable to assume the trend will continue indefinitely?

There is a relative minimum at week 4. As n → ∞, w(n) → ∞ .

Example: The graph models the cross section of Mount Rushmore. What is the smallest degree possible for the equation that corresponds with this graph?