Polynomial FunctionsPower FunctionSpecial Power FunctionsSlide 4Slide 5PolynomialsSlide 7Polynomial PropertiesSlide 9Slide 10Zeros of PolynomialsSlide 12Methods for Finding ZerosPracticeAssignmentPolynomial FunctionsLesson 11.2Power Function•Definition •Recall from the chapter on shifting and stretching, what effect the k will have?Vertical stretch or compression py k x= �for k < 1Special Power Functions•Parabola y = x2•Cubic function y = x3•Hyperbola y = x-1Special Power Functions•y = x-2• • 12y x=133y x x= =Special Power Functions•Most power functions are similar to one of these six •xp with even powers of p are similar to x2•xp with negative odd powers of p are similar to x -1•xp with negative even powers of p are similar to x -2•Which of the functions have symmetry?What kind of symmetry?Polynomials•Definition:The sum of one or more power functionEach power is a non negative integer3 24( ) 5 7 123f x x x x= - + -Polynomials•General formulaa0, a1, … ,an are constant coefficientsn is the degree of the polynomialStandard form is for descending powers of xanxn is said to be the “leading term”11 1 0( ) ...n nn nP x a x a x a x a--= + + + +Polynomial Properties•Consider what happens when x gets very large negative or positiveCalled “end behavior”Also “long-run” behavior•Basically the leading term anxn takes over•Comparef(x) = x3 with g(x) = x3 + x2Look at tablesUse standard zoom, then zoom outPolynomial Properties•Compare tables for low, high valuesPolynomial Properties•Compare graphs ( -10 < x < 10)For 0 < x < 500the graphs are essentially the sameThe leading term x3 takes overThe leading term x3 takes overZeros of Polynomials•We seek values of x for which p(x) = 0•We have the quadratic formula•There is a cubic formula, a quartic formulaZeros of Polynomials•We will use other methods•ConsiderWhat is the end behavior?What is q(0) = ?How does this tell us that we can expect at least two roots?6 5 2( ) 3 2 4 1q x x x x= - + -Methods for Finding Zeros•Graph and ask for x-axis intercepts•Use solve(y1(x)=0,x)•Use zeros(y1(x),x)•When complex roots exist, use cSolve() or cZeros()Practice•Giveny = (x + 4)(2x – 3)(5 – x)What is the degree?How many terms does it have?What is the long run behavior?•f(x) = x3 +x + 1 is invertible (has an inverse)How can you tell?Find f(0.5) and f -1(0.5)Assignment•Lesson 11.2•Page 445•Exercises 1 – 29
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