Linear Functions Answer Key 1. p(x) is a linear function and p(0) = 4 and p(3) = 5, find its equation and sketch its graph. p(x) = 3x 4 2. r(s) is a linear function and r( 2) = 6 and r(3) = 2, find its equation and sketch its graph. r(s) = 4 22 s + 5 5 3. n(t) is a linear function and n(0) = 1 and n(4) = n(0), find its equation and sketch its graph. n(t) = 1
4. a(x) is linear and you know a(0) = 4 and that a(5) a(1) = 3. a(x) = 3 x + 4 4 2 2 5. m(x) is a linear function and m(0) = 2 and m(3) m(1) = 4. m(x) = 2x + 2 6. h(t) is a linear function and h( 2) = 0 and h(1) h(0) = 1 4. h(t) = 1 4 t + 1 2
7. b(t) is a linear function and b( 2) = 6 and b(4) b(1) = 2. b(t) = 2 14 t + 3 3 8. k(x) is a linear function and k(1) = 3 and k(6) k(4) = 1. k(x) = 1 2 x + 5 2 9. g(t) is a linear function and g( 2) = 1 and g(t + 2) g(t) = 5 for any value of t. g(t) = 5 2 t + 6
10. f(x) is a linear function and f(3) = 1 and f(x + 6) f(x) = 2 for any value of x, find its equation and sketch its graph. f(x) = 1 3 x 11. Suppose L(x) = 3x + 2 a) What would L(x + 1) L(x) be for all x? 3 b) What would L(x + 3) L(x) be for all x? 9 c) What would L(x 2) L(x) be for all x? -6 12. a) Consider the graph of k(x) below, does k(0) = 4 represent the length of a line, an area, a slope, a point, or what? Could be either length or point b) On the graph of k(x) below, indicate what k(a + 2) k(a) represents. Is this a point, the length of a line, an area, a slope, or what? Length (change in y) c) What does k(a+2) k(a) 2 represent, a point, the length of a line, an area, a slope, or what? slope 13. a) What is the supply if the booth charges $2? What is the demand? Supply is 600 bags and demand is 3000 bags. b) What happens to supply if the price is increased by $1? Supply increases by 1000 bags. c) What is the equilibrium price? $3.30 d) At what prices will there be a surplus? p > $3.30 e) At the equilibrium price, what is the booth s revenue? $6600 f) If the booth rental costs $700 and ingredients cost $0.30 per bag of cotton candy find the booth s profit at equilibrium price. $5300 g) Find the slope of D and explain its meaning using everyday language. The slope is -800. This tells us that when the price is increased by $1 demand will drop by 800 bags. h) Find an equation for D(p) = 4600 800p i) Determine a reasonable domain and range for D(p) Domain: [0, 5.8] Range: [0, 4600] 14. The following graph represents the distance that Ann and Betty run in a race after a given time. Since Ann is slower she gets a head start. a) How much of a head start does Ann have? Ann has a 25 foot head start. b) How far can each woman run after 5 seconds? Betty can run 50 feet and Ann can run about 33 feet. c) Who wins the race if the race is 60 feet long? Ann will win. d) At what time does Betty overtake Ann? She catches up after 7 seconds. e) What is the longest race Ann could win? Ann can win if the race is less than 70 feet long.
Temperature o F 15. The formula for converting temperature from degrees Celsius to degrees Fahrenheit is F(C) = 9 C + 32. 5 a) If the temperature is 20 o C what is it in o F? 68 F b) If the temperature is 110 o F what is it in o C? 43.33 C c) Find the temperature at which the reading is the same on both scales. -40 d) Find both intercepts and explain their meanings in everyday language. vertical intercept (0, 32): If the temperature is 0 C it is 32 F. horizontal intercept (-17.78, 0): If the temperature is -17.78 C it is 0 F. e) Sketch your model over a reasonable domain. 140 120 100 80-60 40 20 0-30 -10-20 10 30 50-40 Temperature o C 16. It has been found that for every hour spent weightless a person s reaction time increases. After 1 hour average reaction time is 0.35 seconds and after 6 hours it is 1 second. a) Find a model giving a person s reaction time R as a function of t, the number of hours spent weightless. r = reaction time (seconds) w = number of hours spent weightless R(t) = 0.13t + 0.22 b) What would their reaction time be if they had been weightless 2 hours? 0.48 seconds c) How much would reaction time increase if they stayed weightless an additional 4 hours? 0.52 seconds d) If a person had a reaction time of 4 seconds estimate how many hours the person had spent weightless. 29.08 hours e) Find the vertical intercept of your model and explain its meaning in everyday language. (0, 0.22) Without being weightless a person s reaction time would be 0.22 seconds. 17. The expected weight W (in tons) of a humpback whale can be approximated from its length L (in feet) by using W(L) = 1.70L 42.8 for 30 L 50. a) Find and interpret W(40) A whale that is 40 feet long should weigh 25.2 tons. b) Interpret the slope of the model. For every additional 1 foot of length a whale s weight increases by 1.7 tons. 18. Margaret was running at a pace of 5.75 meters per second during a race. After the first minute, the finish line was 455 meters away. KTS a) Find a linear model for meters remaining in the race in terms of time (seconds). d = distance to finish line (meters) t = time since the race started (seconds) d = 5.75t + 800 b) 90 seconds into the race Margaret gets passed, how far from the finish line is she? 282.5 meters c) Interpret both intercepts of your model. vertical intercept: (0, 800) The race is 800 meters long. horizontal intercept: (139.13, 0) It takes Margaret 139.13 seconds to finish the race. d) Find a reasonable domain for your model. 0 t 139.13
19. The amount of heat H (in joules) required to convert one gram of water into vapor is linearly related to temperature T (in o C) of the atmosphere. At 10 o C this conversion requires 2480 joules, and each increase in temperature of 15 o C lowers the amount of heat needed by 40 joules. Find a linear model for H in terms of T. H = 2.67T + 2506.7 20. The owner of an ice cream franchise must pay the parent company $1000 per month plus 5% of the monthly revenue. Operating cost of the franchise includes a fixed cost of $2600 per month for items such as utilities and labor. The cost of ice cream and supplies is 50% of the revenue. a) Find a model giving the owner s monthly expenses as a function of revenue and interpret its slope. E = owner s expenses R = monthly revenue ($) E = 0.55R + 3600 If the revenue increases by $1 the owner s expenses increase by $0.55. b) Find a model giving the monthly profit as a function of revenue. P = monthly profit ($), P = 0.45R 3600 c) Interpret the slope of your profit model. For every additional $1 in revenue profit increases by $0.45. d) Determine the monthly revenue needed to break even. $8000