Math Algebra Pre-Calculus Worksheet Answers

3.1 Worksheet Name:_______________________________________ Hour:________ Honors Pre-Calculus Use your calculator to evaluate the function at the indicated value for x. Round your resolute to three decimal places. 1. 𝑓(𝑥) = 5𝑥 ; 𝑥 = −𝜋 2. 1 𝑓(𝑥) = 𝑒 𝑥 ; 𝑥 = 2 3. 𝑓(𝑥) = 8.6−3𝑥 ; 𝑥 = −√2 Graph the exponential function by hand. Identify the domain and range, any asymptote, any intercepts, and determine whether the graph of the function is increasing or decreasing. 4. 𝑓(𝑥) = 5𝑥 5. 𝑓(𝑥) = 5−𝑥 − 3 6. 𝑓(𝑥) = 5𝑥 − 2 Domain: Domain: Domain: Range: Range: Range: Asymptote(s): Asymptote(s): Asymptote(s): Intercepts: Intercepts: Intercepts: Increasing/Decreasing Increasing/Decreasing Increasing/Decreasing [circle one] [circle one] [circle one] 1 𝑥 7. 𝑓(𝑥) = (2) ; 1 −𝑥 𝑔(𝑥) = 1 − (2) D of 𝑓(𝑥):________________________ R of 𝑓(𝑥):________________________ Asymptote of 𝑓(𝑥):________________ 𝑓(𝑥): Inc. / Dec. _________________ D of 𝑔(𝑥):________________________ R of 𝑔(𝑥):________________________ Asymptote of 𝑔(𝑥):________________ 𝑔(𝑥): Inc. / Dec. _________________ 8. 𝑥 𝑓(𝑥) = 𝑒 ; 𝑔(𝑥) = −2𝑒 𝑥+1 Transformation(s) from 𝑓(𝑥) to 𝑔(𝑥): D of 𝑓(𝑥):________________________ R of 𝑓(𝑥):________________________ Asymptote of 𝑓(𝑥):________________ 𝑓(𝑥): Inc. / Dec. _________________ D of 𝑔(𝑥):________________________ R of 𝑔(𝑥):________________________ Asymptote of 𝑔(𝑥):________________ 𝑔(𝑥): Inc. / Dec. _________________ Transformation(s) from 𝑓(𝑥) to 𝑔(𝑥): 9. Tom makes an investment of $2500 at 2.5% for 10 years. Compare the amount he would have earned in 10 years if the investment was compounded quarterly versus continuously. 10. Christina has $10,000 to invest over a four year period. She makes an appointment to see her financial advisor. She is presented with three options. Which one is best? 11. Plan A: Interest rate of 2.55%, compounded monthly Plan B: Interest rate of 2.75%, compounded semi-annually Plan C: Interest rate of 2.6%, compounded daily 4 The demand function for a product is given by 𝑝 = 5000 [1 − 4 + 𝑒 − 0.002𝑥 ] where 𝑝 is the price, and 𝑥 is the number of units. a. Use a graphing utility to graph the demand function for 𝑥 > 0 and 𝑝 > 0. b. Find the price 𝑝 for a demand of 𝑥 = 500 units. Let 𝑄 represent a mass, in grams, of radioactive radium ( 226𝑅𝑎), whose half-life is 1599 years. 12. 𝑡 The quantity of radium present after 𝑡 years is given by 𝑄 = 1 1599 25 ( ) . 2 a. Determine the initial quantity (when 𝑡 = 0). b. Determine the quantity present after 1000 years. c. Use a graphing utility to graph the function over the interval 𝑡 = 0 to 𝑡 = 5000. d. When will the quantity of radium be 0 grams? Explain. A certain type of bacteria increases according to the model 𝑃(𝑡) = 100𝑒 0.2197𝑡 where 𝑡 is the time in hours. 13. a. Use a graphing utility to graph the model. b. Use a graphing utility to approximate 𝑃(0), 𝑃(5) and 𝑃(10). 14. The projected population of California for the years 2015 to 2030 can be modeled by 𝑃 = 34.706𝑒 0.0097𝑡 where 𝑃 is the population (in millions) and 𝑡 is the time (in years) with 𝑡 = 15 corresponding to 2015. a. Use a graphing utility to graph the function for the years 2015 through 2030. b. According to the model, when will the population of California exceed 50 million? 15. In early 2006, a new Jeep Wrangler Sport Edition had a manufacturer’s suggested retail price 3 𝑡 of $23,970. After 𝑡 years the Jeep’s value is given by 𝑉(𝑡) = 23970 (4) . a. Use a graphing utility to graph the function, and use the table on the calculator to identify the value of the Jeep over a 40 year period. You do not have to make a table yourself, just refer to the table on the calculator and scroll through the table from 𝑡 = 0 to 𝑡 = 40. b. According to the model, when will the jeep have no value? 1. 0.006 2. 1.649 3. 9220.217 4. D: (− ∞, ∞); R: (0, ∞); A: 𝑦 = 0; 𝒚-int: (0, 1); Incr. 5. D: (− ∞, ∞); R: (− 3, ∞); A: 𝑦 = − 3; 𝒚-int: (0, − 2); 𝒙-int: (− 0.683, 0); Decr. 6. D: (− ∞, ∞); R: (0, ∞); A: 𝑦 = 0; 𝒚-int: (0, 1 ); 25 Incr. 7. D of 𝒇(𝒙): (− ∞, ∞); R of 𝒇(𝒙): (0 , ∞); A of 𝒇(𝒙): 𝑦 = 0; 𝑓(𝑥) is Decr. 7. D of 𝒈(𝒙): (− ∞, ∞); R of 𝒈(𝒙): (− ∞, 1); A of 𝒈(𝒙): 𝑦 = 1; 𝑔(𝑥) is Decr. 7. Transformation(s) from 𝑓(𝑥) to 𝑔(𝑥): Reflect over 𝑦-axis, Reflect over 𝑥-axis, Up 1 8. D of 𝒇(𝒙): (− ∞, ∞); R of 𝒇(𝒙): (0 , ∞); A of 𝒇(𝒙): 𝑦 = 0; 𝑓(𝑥) is Incr. 8. D of 𝒈(𝒙): (− ∞, ∞); R of 𝒈(𝒙): (− ∞, 0); A of 𝒈(𝒙): 𝑦 = 0; 𝑔(𝑥) is Decr. 8. Transformation(s) from 𝑓(𝑥) to 𝑔(𝑥): Left 1, Vertical Stretch of 2; Reflect over 𝑥-axis 9. Quarterly investment earns $3207.57; Continuous Investment earns $3210.06. The continuous investments earns $2.49 more. 10. Plan B 11b. $421.12 12a. 25 grams 12b. 16.21 grams 12d. Never, the graph has a horizontal asymptote at 𝑄 = 0 14b. 2037 13b. 100, 300, 900 15b. Never, the graph has a horizontal asymptote at 𝑉 = 0 Graphs: 4. 5. 8. 11. 13. 6. 7. 12. 14. 15.