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6.7/6.8 Analyzing Graphs of Polynomials

6.7/6.8 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?

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6.7/6.8 Analyzing Graphs of Polynomials

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  1. 6.7/6.8 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?

  2. 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.

  3. Zeros, Factors, Solutions, and Intercepts • 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. If f(x) is a polynomial function, then these statements are equivalent.

  4. 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.

  5. 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 funct if the point is lower than all nearby points. Local maximum Local minimum

  6. Writing a Cubic Function • Two points determine a line. • Three points determine a parabola • Four points determine a cubic function Remember:

  7. Given Three Intercepts and a Fourth Point on the Graph Example: Given x-intercepts (-2, 0), (-1, 0), (1, 0) and a fourth point on the graph (0, 2), find the equation of a polynomial with the least degree. 1. Change the x-intercept to factor form. Write the polynomial function in factor form with the leading coefficient as a. 2. Substitute the x and y coordinates of the fourth point into the equation. 3. Solve for a. 4. Substitute a into the equation.

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