5.8 Analyzing Graphs of Polynomials

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5.8 Analyzing Graphs of Polynomials How do you find the local maximum and minimum on a polynomial graph? What is the maximum number of turning points based on the degree of polynomial? How do you find the equation of polynomial of the least degree given the x-intercepts and another point on the graph?

The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n where n > 0, then the equation has at least one root in the set of complex numbers. This means that the degree of polynomial will tell you the number of solutions to look for. Some of the solutions may be repeating solutions.

Zeros, Factors, Solutions, and Intercepts If f(x) is a polynomial function, then these statements are equivalent. Zero: k is a zero of the polynomial. Factor: x – k is a factor of the polynomial. Solution: k is a solution of the polynomial equation f(x). Intercept: If k is a real number then k is an x-intercept of the graph of the polynomial.

Turning Points of Polynomial Functions The graph of every polynomial function of degree n has at most n – 1 turning points. Moreover, if a polynomial function has n distinct real zeros, then its graph has exactly n -1 turning points. Reverse: If you have 2 turning points, then you will have n=3

Local Maximum and Minimum The y-coordinate of a turning point is a local maximum of the function if the point is higher than all nearby points. The y-coordinate of a turning point is a local minimum of the function if the point is lower than all nearby points. Local maximum, 3 Local minimum, -3

Determine the x-intercepts & Graph the polynomial functions.

Determine the x-intercepts & Graph the polynomial functions.