Exp, Log, Poly Functions Quarter 3 Review Name Textbook problems for practice: p. 285-293; p. 293 #9-14, p. 294-5 #1-34, 49-52, 55,56, 57; p. 297-321 logs; p. 280-1 #11-84 *Blood Alcohol, Bungee-from binder (review Pennies, Bacteria, M&M, Population, Prices) Activity Binder *Exponential growth and decay through e x. *Use slopes and successive ratios to identify types of growth. *Write equations from problem situations, graphs. *Compare linear versus exponential growth in terms of their graphs, tables, equations, and the constants associated with them. *Rate of Change, slope, difference quotient *% rate of change, Successive Ratio *Transformations: g(x) = a b (x-h) + k and h(x) = a log b (x-h) +k and k(x) = a f(x-h) + k (any function *State the parent function; track (0,1) & (1,b) or (1,0) and (b,1); graph and identify the transformed functions (no calculator). *Identify the transformations from a given equation; Sketch the graph (no calculator) *Asymptotes, initial values, domain, range. *Slope function on the calculator; successive ratio on calculator. *log(x) and ln(x) rules simplify and evaluate expressions (flash card type) no calc PRACTICE PROBLEMS 1. Graph: y = -½ 3 x-4-2. Track (0,1) & (1,b). State the transformations and asymptote. (no calc.) 2. Write the equation of an exponential function going through the points (1, 3.25) and (4, 8.75) that has an asymptote at y = 1.5. 3. Find the equation of the function: x y asy. at y=-2 0 2 1-2/3 2-14/9 3-50/27 x y -2 15-1 5 0 1 1 3 2 11 3 25
x 2 3 4 5 y -1/3-7/30-49/300-343/3000 4. Find the equation of the slope (remember short cuts and watch for tricks): f(x) = -3(1.2) x g(x) = ⅔ (4) x+2 h(x) = 5(½) x j(x) = 2e x k(x) = - e x 5. State the percent rate of growth or decay for the functions f(x), g(x) and h(x) in #4. 6. Find the rate of change at x = 1 for each of the functions in #4. 7. Given the points (0,30) and (1, 45) a. Find the equation for linear growth. b. Find the equation for exponential growth. What is the percent rate of increase? c. Find when each of the functions will have grown to 100. d. Find when the two functions will have the same rate of change. 8. Given the points (-1,30) and (2, 15) a. Find the equation for linear decay. b. Find the equation for exponential decay. What is the percent rate of decrease? c. Find when each of the functions will reach to 10. d. Find when the two functions will have the same rate of change. 9. Graph: y = - 1 / 3 log 4 (x + 2) +3. Track (1,0) & (b,1). State the transformations and asymptote. (no calc.) 10. State the domain, range and asymptote: a. f(x) = log 3 (x+4) -2 b. g(x) = -2 log 5 (x-5) c. h(x) = 4 (.3) x+1-2 11. State always true, sometimes true (find when) or never true. Explain why. a. ln(x(x + 3) = ln(x) + ln(x + 3) b. ln(2x) = ln (x 2 ) 2 c. log ( x ) = log x d. log -2 (x) = log (x) / log (-2) e. log(4) log(16) = log (¼) x y -3 15-2 11-1 7 0 3 1-1 2-5 3-9 12. Explain the relationship between the domain, range and asymptotes of f(x) = 2 x and g(x) = log 2 (x).
13. Give the domain, range and b values for the function f(x) = log b (x). Explain why the excluded numbers for x and b are not allowed. Answer the following without a calculator. 14. Graph f(x) = log 2 (x). Give the x-intercept and asymptote. How would you describe the 15. Graph f(x) = log 1/2 (x). Give the x-intercept and asymptote. How would you describe the 16. Graph: y = log 3 (x). Give the x-intercept and asymptote. How would you describe the 17. Graph: y = log 1/3 (x). Give the x-intercept and asymptote. How would you describe the 18. Identify the transformations and sketch the graph: y = 3 log (x-5) ½. Where is (1,0) after the transformations? 19. In the chart, state the parent function, f(x), and the transformations for g(x) = ½ log 3 (x-2) + 1 and h(x) = log 2 (3-x) - 4. For each function: Name the new coordinates of the points (1,0) and (b,1) after each transformation. Sketch the graph of g(x). Draw and name the asymptote on the graph. f(x) = (1,0) (, 1) f(x) = (1,0) (, 1) 20. Write as a single logarithm: 4log (x - 3) + ½ log (x) 21. Write as a single logarithm: log (x + 5) log 3 x 22. Give the domain and solve for x algebraically: 3 log 5 (x - 2) = ½. 23. Give the domain and solve for x algebraically: log (x 2) log (2x + 3) = 1 24. Give the domain and solve for x algebraically: log 4 (x + 4) + log 4 (x 4) = 3
25. Give the domain and solve for x algebraically: 3 4ln x = 6 26. Give the domain and solve for x algebraically: ½ log 8 (x + 4) = 1/3 Answer using a calculator: 27. For what values of x is it true that log((x - 2) (x +1)) = log (x - 2) + log (x +1)? Explain. 28. For what values of x is it true that log((x - 3)/ (x +1)) = log (x - 3) log (x +1)? Explain. 29. Explain why log (x 2 ) 2 log (x) for x R. For what values of x is it true? 30. Solve graphically: log (x 2) log (2x + 3) = 1 31. Solve graphically: log 2 (4x 2) log 2 (3x + 1) = 5 32. Solve graphically: 3 log 5 (x-2) = ½ 33. Solve graphically: log 4 (x + 4) + log 4 (x 4) = 3 34. Graph: f(x) = x log (x-1) 2. Give the window, domain, range, roots, asymptotes, max., min. Analyze the graph: ( means approaches) a. As x 1, y?. b. As x +, y?. c. As x -, y?. Polynomial functions: Definition Coefficients, lead coefficient Degree Calculator: Find graphs & windows Roots, extrema, multiplicity Domain & Range FTA EBM, EBT Rational Root Thm Remainder Thm Factor Thm Synthetic division Complex/irrational conjugates Quadratic formula
Determine the remainder when f(x) = 5x 20 3x 7 + 2x 4 5x +1 is divided by 4x 3. Is 4x 3 a factor? Why? Write 2 polynomial functions of degree 6 that have 2 + i, - 4i as non-real roots and 3 and 5 as the only real roots. They may not be a stretch of each other. Use your answers from above to write polynomial functions such that f(0) = 5. Use synthetic division to divide f(x) = 8x 3 60x 2 + 150x 125 by (2x 5). Is (2x 5) a factor? Why? Write f(x) in linear factored form. What are the possible rational roots of g(x) = 8x 5 3x 4 + 7x 2 3x 1? Explain why f(x) = x 5 32 and g(x) = ¼ x 5 8 have the same roots. Use synthetic division to determine if 2 i is a root of f(x) = 2x 3 x 2 + 2x 1. Use the remainder thm to find the remainder when f(x) = x 4 5x 3 +2x 2 6x 5 is divided by (x 4). Write P(x) = x 7 + x 6 5x 5 + 3x 4 25x 3 25x 2 + 125x - 75 in linear factored form using the FTA rational root thm., remainder thm, synthetic division and factor thm. Give the EBM and EBT. Sketch the graph and find the extrema, if any.