7 2 Graphing Polynomial functions Objectives 1 Graph

7 -2 Graphing Polynomial functions Objectives: 1) Graph polynomial functions and locate their real zeros. 2) Find the maxima and minima of polynomial functions.

To graph a polynomial function… • Make a table of values. • Find several points and connect them to make a smooth curve. • Knowing the end behavior of the graph will assist you in completing the sketch of the graph.

Example 1 Graph f(x) = x 4 + x 3 - 4 x 2 - 4 x by making a table of values. x f(x)

Location principle • Notice that the values of the function before and after each zero are a different sign. • In general, the graph of a polynomial function will cross the x-axis somewhere between pairs of x values at which the corresponding f(x) values change sign. • Since zeros of the function are located at the x-intercepts, there is a zero between each pair of these x-values.

Example 2 Determine consecutive values of x between which each real zero of the function f(x) = -5 x 2 + 3 x + 2 is located. Then draw the graph. x f(x)

Maximum and minimum points - Also referred to as TURNING POINTS. - The graph of a polynomial function of degree n has at most n-1 turning points. - Point A on the graph is a relative maximum – no other points nearby have a greater y-coordinate. - Point B is a relative minimum, no other nearby points have a lesser ycoordinate.

Example 3 Graph f(x) = x 3 - 3 x 2 + 5. Estimate the x-coordinates at which the relative maxima and relative minima occur. x f(x)

Example 4 The average fuel (in gallons) consumed by individual vehicles in the United States from 1960 -2000 is modeled by the cubic equation, where t is the number of years since 1960. a) Graph the equation. b) Describe the turning points of the graph and its end behavior. c) What trends in fuel consumption does the graph suggest?

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