6.8 Analyzing Graphs of Polynomial Functions

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6.8 Analyzing Graphs of Polynomial Functions

Zeros, Factors, Solutions, and Intercepts Let f(x) = anxn + an-1xn-1 + … + a1x + a0 be a polynomial function. Zero: k is a zero of the polynomial f(x) Factor: x – k is a factor of the polynomial equation f(x) = 0 Solution: k is a solution of the polynomial equation f(x) If k is a real number, then the following is also equivalent. x-intercept: k is an x-intercept of the graph of the polynomial function f.

Zeros, Factors, Solutions, and Intercepts Graph: f(x) = -2(x2 – 9)(x + 4) Graph: f(x) = x(2x – 5)(x + 5) Graph: f(x) = (x + 3)(x – 4)(x + 1) Graph: f(x) = 2(x – 1)(x + 2)3

Turning Points of Polynomial Functions Local maximum- point higher than all nearby points Local minimum- point lower that all nearby points 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

Examples Graph each function. Identify the x-intercepts, local maximums, and local minimums f(x) = x3 + 2x2 – 5x + 1 f(x) = 2x4 – 5x3 – 4x2 – 6 f(x) = x3 – 5x2 + 4x + 3 f(x) = 3x4 – 4x3 – x2 + 2x – 5

Examples You want to make a rectangular box that is x cm high, (x + 5) cm long, and (10 – x) cm wide. What is the greatest volume possible? What will the dimensions of the box be?

Examples A rectangular box is to be x in. by (12 – x) in. by (15 –x) in. What is the greatest volume possible? What will the dimensions of the box be?