Analyzing Graphs of Polynomials Section 3.2

First a little review… Given the polynomial function of the form: 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 - 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 – 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!

(Think of a line with positive slope!) A n < 0, Odd Degree (Think of a line with negative slope!) A n > 0, Even Degree (Think of a parabola graph… y = x 2.) A n < 0, Even Degree (Think of a parabola graph… y = -x 2.) 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 )  End Behavior A n > 0, Odd Degree y x y x y x y x A n < 0, Odd DegreeA n > 0, Even Degree A n < 0, Even Degree

Examples of End Behaviors

What happens in the middle? The graph “turns” ** 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. Relative maximum Relative minimums ** The lowest degree of a polynomial is (# turning points + 1). So, the lowest degree of this polynomial is 4 !

What’s happening? As x  +, f(x )  As x  -, f(x )  Relative Maximums Relative Minimums The lowest degree of this polynomial is _____. 5 The leading coefficient is __________. positive The number of turning points is _____. 4

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

Example #2: Graph the function: f(x) = x 4 – 4x 3 – x x – 2 and identify the following. End Behavior: _________________________ Degree of polynomial: ______________ # Turning Points: _______________________ y-intercept: _______ Graphing with a calculator Positive-even polynomial of degree 4 As x  -, f(x)  As x  +, f(x)  Plug equation into y= 2.Find minimums and maximums using your calculator Absolute minimum Relative minimum Relative max Real Zeros (0, -2)

Example #3: Graph the function: f(x) = x 3 + 3x 2 – 4x and identify the following. End Behavior: _________________________ Degree of polynomial: ______________ # Turning Points: _______________________ Graphing without a calculator Positive-odd polynomial of degree 3 As x  -, f(x)  As x  +, f(x)  Factor and solve equation to find x-intercepts 2. Plot the zeros. Sketch the end behaviors. f(x)=x(x 2 + 3x – 4) = x(x - 4)(x + 1)

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(a) is negative. (The y-value is negative.) P(b) is positive. (The y-value is positive.) Therefore, there must be at least one real zero in between x = a & x = b! Example #4: Use the Zero Location Theorem to verify that P (x) = 4x 3 - x 2 – 6x + 1 has a zero between a = 0 and b = The graph of P(x) is continuous because P(x) is a polynomial function. 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. (x – c) __________________________________. x = c __________________________________. (c, 0) __________________________________. is a factor of P is a real solution of P(x) = 0 is a real zero of P is an x-intercept of the graph of y = P(x)

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

Assignment: Write the questions and show all work for each. pp #1-13 ODD, 17 & 19 (TI-83), ODD, 33, 35, 41, & 43