LP12 ConnectingRootstoCurveSketching.notebook October 13, 2016

Transcription

1 Warm Up Solve the equation algebraically for all values of x. 1

2 Relationships between Polynomials Equations and their Roots & Signs Case 1: POSITIVE ODD (Meaning the leading coefficient is positive and it is an odd degree.) Case 2: NEGATIVE ODD (Meaning the leading coefficient is negative and it is an odd degree.) Case 3: POSITIVE EVEN (Meaning the leading coefficient is positive and it is an even degree.) + even Case 4: NEGATIVE EVEN (Meaning the leading coefficient is negative and it is an even degree.) 2

3 A zero or root of a polynomial function is the value of x such that f(x)= 0 x = 3 is a root (x+3) is a factor x = 2 is a root (x 2) is a factor x = 4 is a root (x 4) is a factor x = 1 is a root (flat???) (x+1) is a factor x = 1 is a root (x 1) is a factor 3

4 Multiplicity of Roots Even multiplicity roots touch the x axis. Odd multiplicity roots cross the x axis. The larger the multiplicity the flatter the graph is at that root. 4

5 Examples: 1) remember this question from test 2? 2) Find a Polynomial of degree 3 whose zeros are 3, 2, 5. 3) Given the graph shown at the left a) Find the roots b) State the factors c) Write a potential equation 5

6 Construct a polynomial function that might have this graph. 6

7 Connecting Roots to Curve Sketching synonyms of "roots": zeros solutions x-intercepts What is the relationship between ros on a graph and the solutions to an equation? What is the relationship between ros on a graph and factors? 7

8 Example 1: The graph of the polynomial function f(x) = x³ + 4x² + 6x + 4 is shown below. a) Based on the appearance of the graph, what seems to be the real solution to the equation: y = x³ + 4x² + 6x + 4? b) Jiju does not trust the accuracy of the graph. Prove to her algebraically that your answe is in fact a zero of y = f(x). c) Write f as a product of a linear factor, and a quadratic factor, each with realnumber coefficients. d) What is the value of f(10)? Explain how knowing the linear factor of f establishes that f(10) is a multiple of 12. e) Find the two complex number zeros of y = f(x). f) Write f as a product of three linear factors. 8

9 Example 2: Geographers, while sitting at a cafe, discuss their field work site, which is a hill and neighboring river bed. The hill approximately 1050 feet high, 800 feet wide, with peak about 300 feet east of the western base of the hill. The river is about 400 feet wide. They know the river is shallow, no more than about twenty feet deep. They make the following crude sketch on a napkin, placing the profile of the hill and riverbed on a coordinate system with the horizontal axis representing ground level. The geographers do not have with them at the cafe any computing tools, but they nonetheless decide to compute with pen and paper a cubic polynomial that approximates this profile of the hill and riverbed. a) Using only a pencil and paper, write a cubic polynomial function H, that could represent the curve shown (here, x represents the distance, in feet, along the horizontal axis from the western base of the hill, and H(x) is the height in feet of the land at that distance from the western base). Be sure that your formula satisfies H(300) = b) For the sake of ease, the geographers make the assumption that the deepest point of the river is halfway across the river (recall that the river is known to be shallow, with a depth of not more than 20 feet). Under this assumption, would a cubic polynomial provide a suitable model for this hill and riverbed? Explain. 9