Analyzing Graphs of Polynomials Section 3.2

Given the polynomial function of the form: First a little review… Given the polynomial function of the form: f(x) = anxn + an−1xn−1 + . . . + a1x + a0 If k is a zero, Zero: __________ Solution: _________ Factor: _________ If k is a real number, then k is also a(n) __________________. x = k x = k (x – k) x - intercept

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 – must not be a polynomial function A continuous curve is a curve that does not have a break or hole. Hole Break

End Behavior (think a positive slope line!) An < 0 , Odd Degree (think a negative slope line!) An > 0 , Even Degree (think of an x2 parabola graph) An < 0 , Even Degree (think of an -x2 parab. graph) An > 0 , Odd Degree An > 0 , Odd Degree An < 0 , Odd Degree An > 0 , Even Degree An < 0 , Even Degree y x y x y x y x As x  + , f(x) As x  -

What happens in the middle? ** This graph is said to have 3 turning points. Relative maximum The graph “turns” ** The turning points happen when the graph changes direction. This happens at the vertices. ** Vertices are minimums and maximums. Relative minimums The graph “turns” ** The lowest degree of a polynomial is (# turning points + 1). So, the lowest degree of this polynomial is 4 !

What’s happening? 5 Relative Maximums Also called Local Maxes Relative Minimums Also called Local Mins As x  - , f(x) As x  + click The lowest degree of this polynomial is 5 click The leading coefficient is positive

Graphing by hand 2 3 Step 1: Plot the x-intercepts Step 2: End Behavior? Number of Turning Points? Step 3: Plot points in between the x-intercepts. Negative-odd polynomial of degree 3 Example #1: Graph the function: f(x) = -(x + 4)(x + 2)(x - 3) and identify the following. End Behavior: _________________________ # Turning Points: _______________________ Lowest Degree of polynomial: ______________ As x  + , f(x) As x  - , f(x) 2 3 2 Try some points in the middle. (-3, -6), (-1, 12), (1, 30), (2, 24) You can check on your calculator! X-intercepts

Graphing with a calculator Example #2: Graph the function: f(x) = x4 – 4x3 – x2 + 12x – 2 and identify the following. Positive-even polynomial of degree 4 End Behavior: _________________________ # Turning Points: _______________________ Degree of polynomial: ______________ As x  + , f(x) As x  - , f(x) 3 4 Relative max Plug equation into y= Real Zeros Relative minimum Absolute minimum

Graphing without a calculator Example #3: Graph the function: f(x) = x3 + 3x2 – 4x and identify the following. Positive-odd polynomial of degree 3 End Behavior: _________________________ # Turning Points: _______________________ Degree of polynomial: ______________ As x  + , f(x) As x  - , f(x) 2 3 1. Factor and solve equation to find x-intercepts 2. Try some points in the around the Real Zeros Where are the maximums and minimums? (Check on your calculator!)

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 zero (x-intercept) in between x = a & b. a b P(b) is positive. (The y-value is positive.) P(a) is negative. (The y-value is negative.) Therefore, there must be at least one real zero in between a & b!

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