CHAPTER 3 POLYNOMIAL AND RATIONAL FUNCTIONS 3.1 Quadratic Functions вЂў Objectives вЂ“ Recognize characteristics of parabolas вЂ“ Graph parabolas вЂ“ Determine a quadratic functionвЂ™s minimum or maximum value. вЂ“ Solve problems involving a quadratic functionвЂ™s minimum or maximum value. Quadratic functions f(x)= ax пЂ« bx пЂ« c 2 graph to be a parabola. The vertex of the parabolas is at (h,k) and вЂњaвЂќ describes the вЂњsteepnessвЂќ and direction of the parabola given f ( x) пЂЅ a( x пЂ h) пЂ« k 2 Minimum (or maximum) function value for a quadratic occurs at the vertex. вЂў If equation is not in standard form, you may have to complete the square to determine the point (h,k). If parabola opens up, f(x) has a min., if it opens down, f(x) has a max. f ( x) пЂЅ 2 x пЂ 4 x пЂ« 3 2 f ( x) пЂЅ 2( x пЂ 2 x) пЂ« 3 2 f ( x ) пЂЅ 2 ( x пЂ 1) пЂ 2 пЂ« 3 пЂЅ 2 ( x пЂ 1) пЂ« 1 2 2 ( h , k ) пЂЅ (1,1) вЂў This parabola opens up with a вЂњsteepnessвЂќ of 2 and the minimum is at (1,1). (graph on next page) Graph of f ( x ) пЂЅ 2 x пЂ 4 x пЂ« 3 пЂЅ 2 ( x пЂ 1) пЂ« 1 2 2 3.2 Polynomial Functions & Their Graphs вЂў Objectives вЂ“ Identify polynomial functions. вЂ“ Recognize characteristics of graphs of polynomials. вЂ“ Determine end behavior. вЂ“ Use factoring to find zeros of polynomials. вЂ“ Identify zeros & their multiplicities. вЂ“ Use Intermediate Value Theorem. вЂ“ Understand relationship between degree & turning points. вЂ“ Graph polynomial functions. f ( x ) пЂЅ a n x пЂ« a n пЂ1 x n n пЂ1 пЂ« ... пЂ« a 2 x пЂ« a1 x пЂ« a 0 2 вЂў The highest degree in the polynomial is the degree of the polynomial. вЂў The leading coefficient is the coefficient of the highest degreed term. вЂў Even-degree polynomials have both ends opening up or opening down. вЂў Odd-degree polynomials open up on one end and down on the other end. вЂў WHY? (plug in large values for x and see!!) Zeros of polynomials вЂў When f(x) crosses the x-axis. вЂў How can you find them? вЂ“ Let f(x)=0 and solve. вЂ“ Graph f(x) and see where it crosses the x-axis. What if f(x) just touches the x-axis, doesnвЂ™t cross it, then turns back up (or down) again? This indicates f(x) did not change from positive or negative (or vice versa), the zero therefore exists from a square term (or some even power). We say this has a multiplicity of 2 (if squared) or 4 (if raised to the 4th power). Intermediate Value Theorem вЂў If f(x) is positive (above the x-axis) at some point and f(x) is negative (below the x-axis) at another point, f (x) = 0 (on the x-axis) at some point between those 2 pts. вЂў True for any polynomial. Turning points of a polynomial вЂў If a polynomial is of degree вЂњnвЂќ, then it has at most n-1 turning points. вЂў Graph changes direction at a turning point. Graph f ( x ) пЂЅ 2 x пЂ 6 x пЂ« 18 x 3 2 f ( x ) пЂЅ 2 x ( x пЂ 3 x пЂ« 9 ) пЂЅ 2 x ( x пЂ 3) 2 2 Graph, state zeros & end behavior f ( x ) пЂЅ пЂ 2 x пЂ« 12 x пЂ 18 x пЂЅ пЂ 2 x ( x пЂ 6 x пЂ« 9 ) 3 f ( x ) пЂЅ пЂ 2 x ( x пЂ 3) 2 2 2 вЂў END behavior: 3rd degree equation and the leading coefficient is negative, so if x is a negative number such as -1000, f(x) would be the negative of a negative number, which is positive! (f(x) goes UP as you move to the left.) and if x is a large positive number such as 1000, f(x) would be the negative of a large positive number (f(x) goes DOWN as you move to the right.) вЂў ZEROS: x = 0, x = 3 of multiplicity 2 вЂў Graph on next page Graph f(x) f ( x ) пЂЅ пЂ 2 x пЂ« 12 x пЂ 18 x пЂЅ пЂ 2 x ( x пЂ 6 x пЂ« 9 ) 3 f ( x ) пЂЅ пЂ 2 x ( x пЂ 3) 2 2 2 Which function could possibly coincide with this graph? 1) пЂ 7 x пЂ« 5 x пЂ« 1 5 2 )9 x пЂ« 5 x пЂ 7 x пЂ« 1 5 2 3)3 x пЂ« 2 x пЂ« 1 4 2 4) пЂ 4 x пЂ« 2 x пЂ« 1 4 2 3.3 Dividing polynomials; Remainder and Factor Theorems вЂў Objectives вЂ“ Use long division to divide polynomials. вЂ“ Use synthetic division to divide polynomials. вЂ“ Evaluate a polynomials using the Remainder Theorem. вЂ“ Use the Factor Theorem to solve a polynomial equation. How do you divide a polynomial by another polynomial? вЂў Perform long division, as you do with numbers! Remember, division is repeated subtraction, so each time you have a new term, you must SUBTRACT it from the previous term. вЂў Work from left to right, starting with the highest degree term. вЂў Just as with numbers, there may be a remainder left. The divisor may not go into the dividend evenly. Remainders can be useful! вЂў The remainder theorem states: If the polynomial f(x) is divided by (x вЂ“ c), then the remainder is f(c). вЂў If you can quickly divide, this provides a nice alternative to evaluating f(c). Factor Theorem вЂў f(x) is a polynomial, therefore f(c) = 0 if and only if x вЂ“ c is a factor of f(x). вЂў If we know a factor, we know a zero! вЂў If we know a zero, we know a factor!