Chapter 7 Algebra 2 Honors 1 Polynomials Polynomial: - - Polynomials in one variable Degree Leading coefficient f(x) = 3x 3 2x + 4 f(2) = f(t) = f(y -1) = 3f(x) = Using your graphing calculator sketch/graph the following functions on separate graph paper. Note any patterns/facts about the general shape of the graph, noting things such as, but not limited to, the number of turning points, the number of zeros, where the graphs come from and go to at their ends, and how these and other properties relate to properties of the function, such as degree, number of terms, leading coefficient... whew. f(x) = -2x 4 2 f(x) = x 3 4x 2 + 1 f(x) = x 5 x 3 f(x) = -3x + 1 f(x) = -2x 3 + 2x 2 + 1 f(x) = -4x 4 + 4x 2 2 f(x) = x 5 5x 3 + 4x f(x) = - x 5 f(x) = x 5 4x 3 x 4 + 4x 2 f(x) = x 5 x 4 and others that you choose to play with.
Chapter 7 Algebra 2 Honors 2 Page 351 31 37 odd 39-44 Solving polynomial(ish) equations With our Algebra I skills we can solve any equation. With our Algebra II skills we can solve any equation. Beyond these... Find ALL solutions of: x 4 13x 2 + 36 = 0 x 3 + 343 = 0
Chapter 7 Algebra 2 Honors 3 x 6 x 7 Page 363 17, 19, 23, 25, 29 f(x) = 4x 2 3x + 6 divide by x 2 Remainder/Factor Theorems f(2) = Remainder Theorem If you divide f(x) by (x a) then the remainder
Chapter 7 Algebra 2 Honors 4 Synthetic Substitution Factor Theorem If (x a) is a factor then Factoring with Division Factor completely is possible x 4 x 3 8x + 8 if (x 1) happens to be a factor. If (x 2) is a factor of x 2 + kx -17, find k. Page 368 15, 19, 23, 29, 33, 35
Chapter 7 Algebra 2 Honors 5 Finding zeros/roots If f(a) = 0 then a is a of f(x) and (x a) is a of f(x) and if a is a real number then (a, 0) is an of the graph of f(x). A polynomial of degree n has exactly zeros, where these zeros are in the seet of numbers. f(x) = x 5 6x 4 3x 3 + 7x 2 8x + 1 How many zeros does f(x) have? How many of these zeros are real and positive? Real and negative? Complex? Descartes Rule of Signs Page 375 19 23 odd
Chapter 7 Algebra 2 Honors 6 If f(x) has zeros of 3, ¼ and -2, write an equation for f(x). All coefficients must be integers. If f(x) has zeros of 2 and 3 i, write an equation for f(x). All coefficients must be integers. Page 375 37, 39 Rational Zero Theorem Use a graphing calc and find all zeros of each y =10x 3-17x 2 7x + 2 y = 6x 4 + 13x 3 8x 2 17x + 6 List all the possible rational zeros of y = 6x 3 3x 2 + x 8
Chapter 7 Algebra 2 Honors 7 Find all the zeros of f(x) = 2x 4 13x 3 + 23x 2 52x + 60 Page 381 25, 27, 33 Find the domain for the following Domains (revisited) a) f(x) = 3x 2 3x + 2 b) 2x f (x) 4x 1 c) x 1 f (x) x d) f (x) x e) f (x) 3x 2
Chapter 7 Algebra 2 Honors 8 2x 2 x f) f (x) g) f (x) 2 x 1 x 1 Operations on Functions Use these three functions for the following questions: Remember to be on the look out fro Domain issues. f(x) = 4x 2 3x g(x) = 2x + 6 h(x) = 3x 3 + x 4 f(x) + g(x) = (f + g)(x) = g(x) h(x) = (g h)(x) = g(x)f(x) = (f g)(x) = 4g(x) = (h g)(2) = 5f(-1) =
Chapter 7 Algebra 2 Honors 9 g(3)h(-2) = f (3) = g Page 387 19, 20, 21 Compositions f(x) = 4x 2 3x g(x) = 2x + 6 h(x) = 3x 3 + x 4 f(g(x)) = Substitute the function g(x) into the function f(x). This is NOT multiplication. f(g(x)) = g(f(x)) = h(g(x)) = f(h(x)) =
Chapter 7 Algebra 2 Honors 10 f(g(3))= g(h(-2)) = Page 387 30, 34, 37 39, 41, 45 Inverses f -1 (x) is notation for the inverse of the function f(x), s is y -1 if the equation is in y = form. An inverse undoes a relation. In other words, if plugging a 3 into a relation results in a value of 7, then plugging a 7 into the inverse will result in a 3. Find the inverse of f(x) = 3x 4. If 3 f (x) x 2 4, find f -1 (x). If 2x 3 f (x) 6 find f -1 (x)
Chapter 7 Algebra 2 Honors 11 Graph f(x) and f -1 (x) for each of the above examples: Determine whether f(x) = 5x + 10 and g(x) = 1/5x 2 are inverses. In order for the inverse of a function to be a function
Chapter 7 Algebra 2 Honors 12 Select a restricted domain so that the inverse will be a function, then sketch the inverse. Page 393 29, 31, 35, 37 If the following functions consist of the given set of ordered pairs, answer the following: f = (2, 3) (1, 4) (0, 2) (5, 4) g = (1, 2) (3, 4) (5, 0) (2, 5) Is f a function? Find f(g(x)) Find g(f(x))
Chapter 7 Algebra 2 Honors 13 Is f(g(x)) a function? Find f(1)g(1) Find f(2) + g(2) = (f + g)(2) = Find f (3) g Find f -1 (x) Page 387 23, 25, 27 Square Root equations and inequalities Graph: y x y 2x 4
Chapter 7 Algebra 2 Honors 14 y 2 3 x y x 2 State the domain and range for each of the above functions. Graph: y x 1 y 2x 4 Page 398 14 29 column