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Advanced Functions Warm Up Day 8 Name____________ 1. Divide using Synthetic Division. P(x) = 5x3-18x+2 by x-1 2. Divide using long division. x4+3x3-5x+3x2-1 by x-4 3. Use the remainder theorem to find P(c). 4. Use synthetic division and Factor Theorem to determine whether the given binomial is a factor of P(x). P(x) = 3x3+4x2-27x-36, x-4 P(x) = -2x3-2x2-x-20, c=10 Now can you apply what you know to completely factor and solve polynomials? If we know a factor, we can use synthetic division to rewrite the polynomial as a product of the binomial and the reduced polynomial. Then can you factor more? 5. Factor f(x) = 3x3-4x2-28x-16 completely given that x + 2 is a factor. If we know a zero, we can use synthetic division to rewrite the polynomial as a product of the binomial and the reduced polynomial. Then can you find the remaining solutions? 6. Given a polynomial function f and a known zero of f, find all the zeros. f(x) = x3 - 2x 2- 21x - 18; -3 NOTES: 3.2 Graphing Polynomial Functions A polynomial function is a function of the form: f(x) = anxn + an−1xn−1 + . . . + a1x + a0 What are an , an−1 , ..., a1 , a0 ? _______________ What are n, n-1, n-2, … ? ________________ Note: all coefficients must be real numbers and all exponents of variables must be whole numbers!! Example: 2x3 5x2 4x 7 Name the leading coefficient________, the constant_________, and the degree_________, the linear term __________, the quadratic term __________. Give an example of each type of polynomial: Degree Type Constant Standard Form Linear Quadratic Cubic Quartic Nth degree polynomial Some graphing fun. Sketch the general shape of each function: f(x) = 2x f(x) = x3 f(x) = x3 + 1 f(x) = −x3 f(x) = −x3 + 1 Now, repeat: f(x) = −2x What happened to the leading coefficient? What happened to the graph? Sketch again: f(x) = 2x2 f(x) = 2x2 + 3 f(x) = x4 f(x) = −2x2 + 3 f(x) = −x4 Now, repeat: f(x) = −2x2 What happened to the leading coefficient? What happened to the graph? To summarize end behavior… x → +∞ means “as x approaches positive infinity (x gets large)” x → −∞ means “as x approaches negative infinity (x gets small)” an> 0 an< 0 an> 0 an< 0 Odd degree function x → −∞ x → +∞ Even degree function x → −∞ x → +∞ Turning Points: The graph of every polynomial with degree n has at most n – 1 turning points. For example, if the degree is 8, there can be no more than _____ turning points. Alternatively, if there are 7 turning points, the lowest degree will be ____. Vocabulary: Absolute Maximum__________________________________________ Absolute Minimum__________________________________________ Relative Minimum___________________________________________ Relative Maximum___________________________________________ Example: Analyze the graphs. Give the (approximate) coordinates of any local maximums and minimums. State the real zeros. Determine the lowest degree that the function could have. a) b) As x +, f(x)_____ As x +, f(x)_____ As x -, f(x)_____ As x -, f(x)_____ Real Zeros: __________________ Real Zeros: __________________ Relative Maximum(s):___________ Relative Maximum(s):___________ Relative Minimum(s):____________ Relative Minimum(s):____________ Absolute Maximum:_____________ Absolute Maximum:_____________ Absolute Minimum:______________ Absolute Minimum:______________ Number of Turning Points:________ Number of Turning Points:________ Lowest Degree:_______ Lowest Degree:_______ Example: Sketch the graph of f ( x) x( x 2)( x 1)( x 2) . Label x- and y-intercepts, maximum and minimum points, and the graph. x Use a t-chart of values to plot points. Relative maximum(s):__________ Relative minimum(s): __________ Absolute maximum:____________ Absolute minimum:____________ y Roots: __________________ x y 3.2 Homework Advanced Functions Polynomial functions have positive, integer exponents applied to variables. They do not include absolute values, roots, or negative exponents that are applied to variables, and they do not include variables in the denominator. Classify the following functions. Decide if the function is a polynomial function. If it is a polynomial function, state its degree, type, leading coefficient and general shape. 1. f(x) = 2x – ⅔x4 + 9 2. f(x) = x + π 3. f(x) = 3x-2 + 4x-1 + 1 Polynomial?______ Degree:______ L.C.:_______ Type:______________ 4. f ( x) x 2 2 x 5 Polynomial?______ Degree:______ L.C.:_______ Type:______________ 5. f ( x) | x 5 | 3 Polynomial?______ Degree:______ L.C.:_______ Type:______________ 6. f(x) = (x – 5)2 + 3 Polynomial?______ Degree:______ L.C.:_______ Type:______________ 7. f(x) = – x3 + 36x2 – 3x + 7 Polynomial?______ Degree:______ L.C.:_______ Type:______________ 8. f(x) = 25 – 2 Polynomial?______ Degree:______ L.C.:_______ Type:______________ Polynomial?______ Degree:______ L.C.:_______ Type:______________ Polynomial?______ Degree:______ L.C.:_______ Type:______________ Polynomial?______ Degree:______ L.C.:_______ Type:______________ 9. f ( x) 2 x 5 Given the graph, describe the end behavior of the function. Also, state the ordered pairs of the real zeros, the y-intercept, the relative maximum(s) and the relative minimum(s). 10. 11. 12. As x +, f(x) ____ As x -, f(x) ____ Real Zeros:______________ y-intercept:______ Relative maximum(s):_______ Relative Minimum(s):_______ Absolute Maximum:_________ Absolute Minimum:_________ As x +, f(x) ____ As x -, f(x) ____ Real Zeros:______________ y-intercept:______ Relative maximum(s):_______ Relative Minimum(s):_______ Absolute Maximum:_________ Absolute Minimum:_________ Given the function, describe the end behavior. As x +, f(x) ____ As x -, f(x) ____ Real Zeros:______________ y-intercept:______ Relative maximum(s):_______ Relative Minimum(s):_______ Absolute Maximum:_________ Absolute Minimum:_________ 13. f(x) = -x3 + 1 14. f(x) = x5 + 2x2 15. f(x) = 3x8 – 4x3 16. f(x) = -x6 + 2x3 – x As x +, f(x) ____ As x -, f(x) ____ As x +, f(x) ____ As x -, f(x) ____ As x +, f(x) ____ As x -, f(x) ____ As x +, f(x) ____ As x -, f(x) ____ Given the graph, what is the lowest degree that the function could have? 17. 18. 19. Number of turning points:____ Lowest Degree:______ Real Zeros:______________ y-intercept:______ Relative maximum(s):_______ Relative Minimum(s):_______ Absolute Maximum:_________ Absolute Minimum:_________ As x +, f(x) ____ As x -, f(x) ____ Number of turning points:____ Lowest Degree:______ Real Zeros:______________ y-intercept:______ Relative maximum(s):_______ Relative Minimum(s):_______ Absolute Maximum:_________ Absolute Minimum:_________ As x +, f(x) ____ As x -, f(x) ____ Number of turning points:____ Lowest Degree:______ Real Zeros:______________ y-intercept:______ Relative maximum(s):_______ Relative Minimum(s):_______ Absolute Maximum:_________ Absolute Minimum:_________ As x +, f(x) ____ As x -, f(x) ____