## Document Summary

M A T 171 P re-C alcu lu s A lgeb ra C h ap ter 4 Lab N am e: Task 1: Exploration ?- Com plete prior to Section 4.1-W orth 10 points A ?General Form of a Polynom ial Function? can be given by the equation ? w here a, and c are real num bers and . View the follow ing interactive w ebsite. = a /0 https://w w w .desm os.com /calculator/3ojkyjajt2? M ove the sliders a, n, and c and observe how the graph changes. Then ?answ er the follow ing questions. a. Set c=-1 and “a” to a positive num ber. Drag the slider for “n”. Describe w hat happens to the right side of the graph as x approaches infinity regardless of w hether n is even or odd. Describe w hat happens to the left side of the graph as x approaches negative infinity w hen n is even. Describe w hat happens to the left side of the graph as x approaches negative infinity w hen n is odd. W rite your answ er in com plete sentences. b. Set c=-1 and “a” to a negative num ber. Drag the slider for “n”. Describe w hat happens to the right side of the graph as x approaches infinity regardless of w hether n is even or odd. Describe w hat happens to the left side of the graph as x approaches negative infinity w hen n is even. Describe w hat happens to the left side of the graph as x approaches negative infinity w hen n is odd. W rite your answ er in com plete sentences. c. Set “a” and “n” to any num ber. Drag the slider for “c”. Describe how the graph changes as c changes. W hat does “c” represent on the graph? W rite your answ er in com plete sentences. 1 Task 2: Linear, Q uadratic, and Cubic Regression ?- Com plete w ith Section 4.1 - W orth 26 points - each part w orth 1 point unless otherw ise noted. Finding the best m odel: The ?coefficient of determ ination, r?2? tells you how w ell the m odel describes the data. It takes values betw een 0 and 1 and the closer it is to 1, the better the data fits the regression curve. R?2? can be calculated for any type of regression m odel (linear, quadratic, cubic, exponential, etc.). You w ill need technology to do this task. You can use your graphing calculator, excel, or a dem os site. Use the technology you are m ost com fortable w ith. Below are som e resources for each. ? TI 83/T84 Graphic Calculator?- Here is a handout on how to do ?scatter plots and regression? w ith the graphic calculator. This handout is also posted in M yCourses as w ell. Here is a link to a tutorial video as w ell. ?https://w w w .youtube.com /w atch?v=49W 01QLhKw 4 ? Excel? - Here is a handout on how to do ?scatter plots and regression? ?w ith excel. Here is a link to a tutorial video as w ell. ?https://w w w .youtube.com /w atch?v=x3k0U-cxsYA ? Desm os? - Here is a link to dem os online ?https://w w w .desm os.com /calculator/sxvk5lay5b Consider the follow ing: The graph below represents the Foreign-Born Population and Percentage of Total Population from 1850 to 2010. (Source: ?w w w .census.gov?). a. In w hat year w as the largest percentage of of the US population born abroad? b. In w hat year w as the greatest num ber of the US population born abroad? c. Looking at your answ er for “a” and “b”, w hy are they not the sam e year? Explain in com plete sentences. (Hint: it has som ething to do w ith percentages) 2 Let’s look at w hat happened betw een 1850 and 1900. Fill in the table below . W orth 2 points. Table 1. (1850-1900) Year t (Let t represents the num ber of years after 1850) P(t) is the Foreign-Born Population in M illions 1850 0 2.2 1860 1870 1880 1890 1900 ** W hen you go to calculate your regression equations use the tw o colum ns of the data you com pleted above. d. Looking at t and P(t) in Table 1 and the scatterplot, does the data appear to be linear, quadratic, or cubic? e. Using your graphing calculator, Excel, or Desm os, find the linear regression equation P(t) that best fits the data w here t represents the num ber of years since 1850. Also find the coefficient of determ ination, r?2?, rounding to three places. W rite the equation and r?2 ? below . W orth 2 points. f. Using your graphing calculator, Excel, or Desm os, find the quadratic regression equation P(t) that best fits the data w here t represents the num ber of years since 1850. Also find the coefficient of determ ination, r?2?, rounding to three places. W rite the equation and r?2 ? below . Ask yourself is this really a quadratic equation. W orth 2 points. g. Using your graphing calculator, Excel, or Desm os, find the cubic regression equation P(t) that best fits the data w here t represents the num ber of years since 1850. Also find the coefficient of determ ination, r?2?, rounding to six places. W rite the equation and r?2 ? below . Note that if your calculator outputs som ething like 3.459E-4 it is in scientific notation and is equivalent to 0.0003459 as the decim al place m ust be m oved 4 places to the left. W orth 2 points. h. W hat do you notice about your answ ers in parts “e”, “f” and “g”. W hich regression m odel (linear, quadratic, or cubic) best fits in the data in Table 1? 3 i. Using the best fit regression m odel identified in part “h”, predict the foreign born population in 1885. Round 1 decim al place. Rem em ber this answ er represents the foreign born population in ?m illions?. W hat value are you going to plug in for t if t represents the num ber of years since 1850? Now let’s look at w hat happened betw een 1850 and 2010. Fill in the table below . W orth 2 points. Table 2. (1850-2010) Year t (Let t represents the num ber of years after 1850) Foreign-Born Population in M illions 1850 0 2.2 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 ** W hen you go to calculate your regression equations use the tw o colum ns of the data you com pleted above. j. Looking at the t and P(t) in Table 2 and the scatter plot, does the data appear to be linear, quadratic, or cubic? 4 k. Using your graphing calculator, Excel, or Desm os, find the linear regression equation P(t) that best fits the data w here t represents the num ber of years since 1850. Also find the coefficient of determ ination, r?2?, rounding to three places. W rite the equation and r?2 ? below . W orth 2 points. l. Using your graphing calculator, Excel, or Desm os, find the quadratic regression equation P(t) that best fits the data w here t represents the num ber of years since 1850. Also find the coefficient of determ ination, r?2?, rounding to three places. W rite the equation and r?2 ? below . W orth 2 points. m . Using your graphing calculator, Excel, or Desm os, find the cubic regression equation P(t) that best fits the data w here t represents the num ber of years since 1850. Also find the coefficient of determ ination, r?2?, rounding to six places. W rite the equation and r?2 ? below . W orth 2 points. n. Com pare the coefficient of determ ination, r?2?, you calculated in parts “k”, “l”,and “m ”. W hich regression m odel (linear, quadratic, or cubic) best fits in the data in Table 2? o. Using the best fit regression m odel identified in part “n”, predict the foreign born population in 2015. Round 1 decim al place. Rem em ber this answ er represents the foreign born population in ?m illions?. W hat value are you going to plug in for t if t represents the num ber of years since 1850? p. Notice that over tim e, the best fit m odel has changed. W hat is one reason w hy you think the foreign born population has fluctuated such? 5 Task 3: G raphical A nalysis of R ational Functions ?Com plete with Section 4.2 W orth 12 pts Read pages 191193 in your textbook or ebook in M LP prior to com pleting this task. If you need additional help w ith this task visit http://www.coolm ath.com /precalculusreviewcalculusintro/precalculusalgebra/21rationalfunctionslim itsinfinit yrightleft01?. I. Below is the graph of ?. U se the graph to com plete each statem ent. a. A s _____ , (x) x ? ? f ? b. A s _____ ? , (x) x ? ? f ? c. A s _____ ? , (x) x ? 2+ f ? d. A s _____ ? , (x) x ? 2? f ? e. A s _____ , (x) x ? 2+ f ? f. A s _____ , (x) x ? 2? f ? II. Below is the graph of ?. U se the graph to com plete each statem ent. a. A s _____ , (x) x ? ? f ? b. A s _____ ? , (x) x ? ? f ? c. A s _____ ? , (x) x ? 3+ f ? d. A s _____ ? , (x) x ? 3? f ? e. A s _____ , (x) x ? 4+ f ? f. A s _____ , (x) x ? 4? f ? The idea of the lim it of a function is what connects algebra and geom etry to the m athem atics of Calculus. Task 3 allows us in PreCalculus to preview the idea of a lim it by studying the behavior of a rational function as it approaches the asym ptotes. 6 Task 4: G raphical A nalysis of R ational Functions ?Com plete with Section 4.3 W orth 14 pts each part worth 1 point except h is worth 3 points. Consider the function: ? a. W hat is the dom ain of ? (x) f b. State any hole(s) as ordered pairs. If no holes, state NONE. c. State the ?equation? of any vertical asym ptote(s). If none, state NONE. d. State the ?equation? of any horizontal asym ptote(s). If none, state NONE. e. State the ?equation? of any slant/oblique asym ptote(s). If none, state NONE. f. W hat are the y-intercept(s) if any of ? If none, state NONE. W rite the y-intercept as an ordered pair. (x) f g. W hat are the x-intercept(s) if any of ? If none, state NONE. W rite the x-intercept(s) as ordered (x) f pair(s). h. Graph . Sketch ALL asym ptotes on graph as dashed lines. (x) f i. W hat interval(s) on the dom ain is increasing if any? If none, state NONE. (x) f j. W hat interval(s) on the dom ain is decreasing if any? If none, state NONE. (x) f k. W hat interval(s) on the dom ain is concave up if any? If none, state NONE. (x) f 7 l. W hat interval(s) on the dom ain is concave dow n if any? If none, state NONE. (x) f Task 5: G raphical A nalysis of R ational Functions ?Com plete with Section 4.3 W orth 14 ptseach part worth 1 point except h is worth 3 points. Consider the function: ? ?. a. W hat is the dom ain of ? (x) f b. State any hole(s) as ordered pairs. If no holes, state NONE. c. State the ?equation? of any vertical asym ptote(s). If none, state NONE. d. State the ?equation? of any horizontal asym ptote(s). If none, state NONE. e. State the ?equation? of any slant/oblique asym ptote(s). If none, state NONE. f. W hat are the y-intercept(s) if any of ? If none, state NONE. W rite the y-intercept as an ordered pair. (x) f g. W hat are the x-intercept(s) if any of ? If none, state NONE. W rite the x-intercept(s) as ordered (x) f pair(s). h. Graph . Sketch ALL asym ptotes on graph as dashed lines. (x) f i. W hat interval(s) on the dom ain is increasing if any? If none, state NONE. (x) f j. W hat interval(s) on the dom ain is decreasing if any? If none, state NONE. (x) f 8 k. W hat interval(s) on the dom ain is concave up if any? If none, state NONE. (x) f l. W hat interval(s) on the dom ain is concave dow n if any? If none, state NONE. (x) f Task 6: G raphical A nalysis of Polynom ial Function ?Com plete with Section 4.6 W orth 24 pts each part worth 1 pt unless otherwise noted. Consider the equation: ? x x x x4? 5 3+ 7 2? 5 + 6 = 0 a. How m any zeros/solutions does function have? b. Using the Rational Zero Theorem , list all of the possible real rational zeros of the function. c. Use Descartes’ Rule of Signs to determ ine how m any positive real zeros the function m ay have. d. Use Descartes’ Rule of Signs to determ ine how m any negative real zeros the function m ay have. e. Find ALL of the zeros (real & im aginary) of the polynom ial equation. GIVE EXACT ANSW ERS. W orth 5 pts ? Use the inform ation gathered in parts a-d, synthetic division, factoring, quadratic form ula, and/or possibly the graph of the polynom ial function as show n by a graphing calculator or Desm os as an aid in obtaining the first zero. ? Verify any zeros obtained from the graphing calculator or Desm os w ith synthetic division. Show ALL synthetic division w ork. 9 f. W hat is the m axim um num ber of turning points of the function? g. Determ ine the end behavior of the graph of the function. h. W hat is the y-intercept of the function? W rite the x-intercept(s) as ordered pair. i. W hat are the x-intercept(s) if any of the function? If none, state NONE. W rite the x-intercept(s) as ordered pair(s). j. W hat is the dom ain of the function? k. W hat is the range of the function? l. Is the function even, odd, or neither? m . Use a graphing calculator* to find the absolute m inim um of the graph. W here does the m inim um occur? W hat is the local m axim um ? Round 1 decim al place. n. Use the inform ation obtained above, to sketch the graph of the function. Be sure to include x-intercepts, y-intercepts, and m axim um /m inim um s on the graph. If necessary, plot additional points. W orth 5 pts o. W hat interval(s) on the dom ain is the function increasing if any? If none, state NONE. p. W hat interval(s) on the dom ain is the function decreasing if any? If none, state NONE. 10 *In Calculus, w e learn how to algebraically find m axim um and m inim um s of functions using the first and second derivative test. Hope you join us in M AT 271 :) 11