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Analyzing Graphs of Polynomials Section 6.8 First a little review… Given the polynomial function of the form: f (x) an xn an1xn1 a1x a0 If k is a zero, x = k Zero: __________ x = k Solution: _________ (x – k) Factor: _________ x - intercept If k is a real number, then k is also a(n) __________________. What kind of curve? All polynomials have graphs that are smooth continuous curves. A smooth curve is a curve that does not have sharp corners. Sharp corner – This graph must not be a polynomial function. A continuous curve is a curve that does not have a break or hole. Hole Break This is not a continuous curve! End Behavior An > 0 , Odd Degree (Think of a line with positive slope!) An < 0 , Odd Degree (Think of a line with negative slope!) y y x An > 0 , Even Degree (Think of a parabola graph… y = x2 .) y y x An < 0 , Even Degree (Think of a parabola graph… y = -x2 .) x x As x - , f(x) As x - , f(x) As x - , f(x) As x - , f(x) As x + , f(x) As x + , f(x) As x + , f(x) As x + , f(x) Examples of End Behaviors x , f ( x) 1. 2. x , f ( x) x , f ( x) x , f ( x) 3. x , f ( x) x , f ( x) x , f ( x) 4. x , f ( x) What happens in the middle? ** This graph is said to have 3 turning points. ** The turning points happen when the graph changes direction. This happens at the vertices. ** Vertices are minimums and maximums. ** The lowest degree of a polynomial is (# turning points + 1). So, the lowest degree of this P( x) x 4 .... polynomial is 4! What’s happening? Relative Maximums Relative Minimums As x - , f(x) As x + , f(x) P( x) x5 .... 4 . The number of turning points is _____ The lowest degree of this polynomial is _____ 5 . positive The leading coefficient is __________ . Graphing by hand Step 1: Plot the x-intercepts Step 2: End Behavior? Number of Turning Points? Step 3: Check in Calculator!!! Negative-odd polynomial of degree 3 ( -x * x * x) f(x) = -(x + 4)(x + 2)(x - 3) Example #1: Graph the function: and identify the following. , f(x) As x + , f(x) As x - End Behavior: _________________________ 3 Lowest Degree of polynomial: ______________ 2 # Turning Points: _______________________ You can check on your calculator!! x-intercepts Graphing with a calculator Example #2: Graph the function: and identify the following. f(x) = x4 – 4x3 – x2 + 12x – 2 As x - , f(x) As x + , f(x) End Behavior: _________________________ 4 Degree of polynomial: ______________ 3 # Turning Points: _______________________ (0, -2) y-intercept: _______ Relative max 1. Plug equation into y= Real Zeros 2. Find minimums and maximums using your calculator Relative minimum Absolute minimum Positive-even polynomial of degree 4 Graphing without a calculator Example #3: Graph the function: and identify the following. f(x) = x3 - 3x2 – 4x As x + End Behavior: _________________________ As x - , f(x) , f(x) 3 Degree of polynomial: ______________ 2 # Turning Points: _______________________ 1. Factor and solve equation to find x-intercepts f(x)=x(x2 - 3x – 4) = x(x - 4)(x + 1) 2. Plot the zeros. Sketch the end behaviors. Positive-odd polynomial of degree 3 Zero Location Theorem Given a function, P(x) and a & b are real numbers. If P(a) and P(b) have opposite signs, then there is at least one real number c between a and b such that P(c) = 0. a b P(b) is positive. (The y-value is positive.) Therefore, there must be at least P(a) is negative. (The y-value is negative.) one real zero in between x = a & x = b! Example #4: Use the Zero Location Theorem to verify that P(x) = 4x3 - x2 – 6x + 1 has a zero between a = 0 and b = 1. The graph of P(x) is continuous because P(x) is a polynomial function. 0 4 -1 -6 1 0 0 0 4 -1 -6 1 1 4 -1 -6 1 4 3 -3 4 3 -3 -2 P(0)= 1 and P(1) = -2 Furthermore, -2 < 0 < 1 The Zero Location Theorem indicates there is a real zero between 0 and 1! Polynomial Functions: Real Zeros, Graphs, and Factors (x – c) • If P is a polynomial function and c is a real root, then each of the following is equivalent. is a factor of P • (x – c) __________________________________ . • x=c is a real solution of P(x) = 0 __________________________________ . • x=c is a real zero of P __________________________________ . an x-intercept of the graph of y = P(x) . • (c, 0) is __________________________________ Even & Odd Powers of (x – c) The exponent of the factor tells if that zero crosses over the x-axis or is a vertex. If the exponent of the factor is ODD, then the graph CROSSES the x-axis. If the exponent of the factor is EVEN, then the zero is a VERTEX. Try it. Graph y = (x + 3)(x – 4)2 Try it. Graph y = (x + 6)4 (x + 3)3