WARM UP EXERCSE. 2-1 Polynomials and Rational Functions

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1 WARM UP EXERCSE Roots, zeros, and x-intercepts. x 2! 25 x x 3! 25x polynomial, f (a) = 0! (x - a)g(x) Polynomials and Rational Functions Students will learn about: Polynomial functions Behavior & graphs Root approximation Rational functions: Behavior & graphs 2

2 (x! 3) = Examples 3 f ( x) = x! 27x f ( x) = x! 6x 4 2 f x x x x 5 3 ( ) =! Behavior as x gets big? How many intercepts? How many turning points? a n x n + a n!1 x n! a 2 x 2 + a 1 x 1 + a 0 3 Graphs of examples f(x)=3x^2+6x-1 y=x^4-6x^ What happens if we shift these? Behavior as x gets big? Behavior as x goes to negative infinity? Number of intercepts? Number of turning points? 4

3 Graphs of examples y=x^3-27x y=x^5-5x^3+4x What happens if we shift these? Behavior as x gets big? Behavior as x goes to negative infinity? Number of intercepts? Number of turning points? 5 Examples x 3! 2 h(x) = (x) 2 (x! 1) g(x) = (x! 1)(x! 2)(x! 3) j(x) = (x! 1)(x 2 + 1) Behavior as x goes to infinity? Behavior as x gets to negative infinity? How many intercepts? How many turning points? 6

4 In General a n x n + a n!1 x n! a 2 x 2 + a 1 x 1 + a 0 n EVEN: Behavior as x goes to infinity? Behavior as x gets to negative infinity? How many intercepts? Between 0 and n How many turning points? Between 1 and n-1 Continuous? n Odd: Behavior as x goes to infinity? Behavior as x gets to negative infinity? How many intercepts? How many turning points? a n > 0 a n < 0 " " "# # f (c) = f (x) x +!c x "!c a n > 0 a n < 0 # 7 Finding the roots (or zeros) of a polynomial If r is a root (or zero) of the polynomial P(x) this means that P(r) = 0. For example, 2 is a second degree polynomial, and p( x) = x! 4x p(4) = so r = 4 is a zero of the polynomial as well as (4,0) being an x-intercept of the graph of p(x). However for other examples it is not so easy and we do not have a version of the quadratic formula for n> f ( x) = x! 5x + 4x + 1 Luckily there is a Theorem (by Cauchy) to help us out. In the example above it says that any root r of f(x) must satisfy the condition that: r < 1+ max{1,5,4,1} This significantly its our search! 8

5 Cauchy s Theorem approximating roots: Given a polynomial x n + a n!1 x n! a 2 x 2 + a 1 x 1 + a 0 If r is a root of f(x) then r < 1+ max{,,...,, } Thus we know that all possible x intercepts (roots) are found along the x-axis between and. So we know now where to look for the zeros. For example we could set our viewing rectangle on our calculator to this window and graph the polynomial function. 9 Example: Approximate the real zeros of Another Example: P x x x x Q(x) = 3 2 ( ) = r < 1+ max{,,, } Roots of new polynomial are the same as the roots of P(x). We know that all possible x intercepts (roots) are found along the x-axis between and. So we set our viewing rectangle on our calculator to this window and graph the polynomial function. Plug and chug to see that the root is approximately (there is only one root). 10

6 Rational Function Examples Graph the following: 1 x Domain Range # x +!2 x "!2 11 Rational Function Examples Graph the following: g(x) = 1 x + 3 = Domain Range # Remark: Limits to infinity in general: ax + b cx + b = ax cx + b + b cx + b = How about as x approaches 0? g(1 /10) = g(1 /100) = g(!1 /10) = g(!1 /100) = as x approaches 0 from the right f (x) approaches x +!0 as x approaches 0 from the left f (x) approaches x "!0 12

7 Rational Function Examples Graph the following: 1 x! 2 Domain Range # How about as x approaches 2? f (1.9) = f (1.99) f (2.1) = f (2.01) x +!2 x "!2 13 Rational Function Examples Graph the following: 1 g(x) = 2(x! 1) + 3 = Domain Range # x +! x "! 14

8 Rational Functions Definition: A Rational function is a quotient of two polynomials, P(x) and Q(x): R(x)=P(x)/Q(x). Example: Let P(x) = x + 5 and Q(x) = x 2 then R(x)= Domain: x + 5 x! 2 Range: Zeros: x-intercepts: y-intercepts: 15 Graph of rational function 16

9 Graph of a Rational function:! Plot points near the value at which the function is undefined. In our case, that would be near x = 2. Plot values such as 1.5, and 2.1, 2.3, 2.5.! Determine what happens to the graph of f(x) if x increases or decreases without bound. That is, for x approaching positive infinity or x approaching negative infinity.! Sketch a graph of a function through these points. 17 Zeros: Rational Functions Definition: A Rational function is a quotient of two polynomials, P(x) and Q(x): R(x)=P(x)/Q(x). We will focus on R(x)= ax + b cx + d Domain: Don t want cx+d = 0. So All real numbers except x=-d/c. The line x=-d/c is the vertical aspmptote Range: a / c # a / c y=a/c is the horizontal asymptote Range: All real numbers except y=a/c. Want ax+b=0 so zero at x=-b/a (if a not zero) x-intercepts: (-b/a,0) y-intercepts: (0,b/d) x +! x "! 18

10 3x + 5 x + 1 Domain: Example: Range: # Zeros: x-intercepts: y-intercepts: x +! x "! 19