7-35 (Odd-numbered;pp.231-232)

SECTION 2.5 = Linear Functions and Models 231 Mary is traveling faster than John. We can see this from the graph because Mary's line is steeper (has a greater slope) than John's line. (b) Let f(x) be the distance John has traveled at time x. Since the speed (average rate of change) is constant, it follows that f is a linear function. Thus we can write in the form f(x) = ax + b. From part (a) we know that the slope a is 50, and from the graph we see that the y-intercept b is 150. Thus the distance that John has traveled at time x is modeled by the linear function f(x) = 50x + 150 Model for John's distance Similarly, Mary is traveling at 75 mi/h, and the y-intercept of her graph is 0. Thus the distance she has traveled at time x is modeled by the linear function 9 (x ) = 75x Model for Mary's distance (c) Replacing x by 5 in the models that we obtained in part (b), we find that at 5:00 P.M. John has traveled f(5) = 50(5) + 150 = 400 mi and Mary has trav- eled g(5) = 75(5) = 375 mi. 500 (d) Mary overtakes John at the time when each has traveled the same distance, that 400 John is, at the time x when f(x) = g(x). So we must solve the equation 300 50x + 150 = 75x John's distance = Mary's distance 200 Mary 100 Solving this equation, we get x = 6. So Mary overtakes John after 6 h, that is, at 6:00 P.M. We can confirm our solution graphically by drawing the graphs of f and 5 6 g on a larger domain as shown in Figure 6. The graphs intersect when x = 6. From the graph we see that the graph of Mary's trip is below the graph of John's FIGURE 6 John and Mary's trips trip from x = 0 to x = 6, so Mary is behind John from noon until 6:00 P.M. Now Try Exercise 45 2.5 EXERCISES CONCEPTS 3. What is the slope of the graph? 1. Let f be a function with constant rate of change. Then 4. At what rate is the pool being filled? (a) f is a function and f is of the form 5. If a linear function has positive rate of change, does its graph f (x) =. slope upward or downward? (b) The graph of f is a 6. Is f(x) = 3 a linear function? If so, what are the slope and 2. Let f be the linear function f(x) = -5x + 7. the rate of change? (a) The rate of change of f is (b) The graph of f is a - with slope and SKILLS y-intercept 7-14 - Identifying Linear Functions Determine whether the 3-4 A swimming pool is being filled. The graph shows the given function is linear. If the function is linear, express the func- number of gallons y in the pool after x minutes. tion in the form f(x) = ax + b. *. 7. f ( x) = 3+ 3x 8. f (x) = 2 - 4x 9 . f ( x ) = x (4 - x) 10. f(x) = Vx + 1 11. f(x) = x+ 1 5 12 . f ( x ) = - 2x - 3 Volume of water (gal) 13 . f ( x ) = (x + 1 ) 2 14. f (x) = 2(3x - 1) 15-18 Graphing Linear Functions For the given linear func- tion, make a table of values and sketch its graph. What is the 10 slope of the graph? 15. f (x) = 2x - 5 16. g(x) = 4 - 2x Time (min) 17. r(t ) = - 3t+ 2 18. h(t) = 2 - at232 CHAPTER 2 . Functions 19-26 = Slope and Rate of Change A linear function is given. increasing the value of a affect the graph of f? What about (a) Sketch the graph. (b) Find the slope of the graph. (c) Find the the rate of change of f? rate of change of the function. 38. Families of Linear Functions Graph f(x) = x + b for 19. f(x) = 2x - 6 20. g(z) = -32 - 9 b = 1. b = 1, and b = 2, all on the same set of axes. How 21. h(t) = -0.51 - 2 22. s(w) = -0.2w - 6 does increasing the value of b affect the graph of f? What about the rate of change of f? 23. U(1) = -101 - 20 24. A(r) = -3r - 1 25. f( 1 ) =-71+ 2 26. g(x) = Ex - 10 APPLICATIONS 27-30 = Linear Functions Given Verbally A verbal description .39. Landfill The amount of trash in a county landfill is modeled of a linear function f is given. Express the function f in the form by the function f (x) = ax + b. T(x) = 150x + 32,000 27. The linear function f has rate of change 3 and initial value - 1 where x is the number of years since 1996 and T(x) is mea- 28. The linear function g has rate of change - 12 and initial sured in thousands of tons. value 100. (a) Sketch a graph of T. 29. The graph of the linear function h has slope , and y-intercept 3. (b) What is the slope of the graph? 30. The graph of the linear function & has slope - and c) At what rate is the amount of trash in the landfill increase y-intercept -2. ing per year? 40. Copper Mining The amount of copper ore produced from a 31-32 - Linear Functions Given Numerically A table of values copper mine in Arizona is modeled by the function for a linear function f is given. (a) Find the rate of change of f. (b) Express f in the form f(x) = ax + b f(x) = 200 + 32x 31. 32. where x is the number of years since 2005 and f(x) is mea- X f(x) x f(x) sured in thousands of tons. 0 - 3 11 (a) Sketch a graph of f. 2 10 2 (b) What is the slope of the graph? 13 - 4 (c) At what rate is the amount of ore produced changing? NUNC 16 - 13 19 - 19 41. Weather Balloon Weather balloons are filled with hydrogen and released at various sites to measure and transmit data about conditions such as air pressure and temperature. A 33-36 - Linear Functions Given Graphically The graph of a weather balloon is filled with hydrogen at the rate of 0.5 ft/s. linear function f is given. (a) Find the rate of change of f. Initially, the balloon contains 2 fts of hydrogen. (b) Express f in the form f(x) = ax + b. (a) Find a linear function V that models the volume of 33. 34. hydrogen in the balloon at any time t. b) If the balloon has a capacity of 15 ft', how long does it take to completely fill the balloon? 42. Filling a Pond A large koi pond is filled from a garden hose at the rate of 10 gal/min. Initially, the pond contains 300 gal of water. (a) Find a linear function V that models the volume of water 5 x in the pond at any time t. 35. (b) If the pond has a capacity of 1300 gal, how long does it take to completely fill the pond? 43. Wheelchair Ramp A local diner must build a wheelchair ramp to provide handicap access to the restaurant. Federal building codes require that a wheelchair ramp must have a maximum rise of 1 in. for every horizontal distance of 12 in. 0 (a) What is the maximum allowable slope for a wheelchair ramp? Assuming that the ramp has maximum rise, find a linear function H that models the height of the ramp above the ground as a function of the horizontal distance x. (b) If the space available to build a ramp is 150 in. wide, SKILLS Plus how high does the ramp reach? 37. Families of Linear Functions Graph f(x) = ax for a = 2, 44. Mountain Biking Meilin and Brianna are avid mountain a = 1, and a = 2, all on the same set of axes. How does bikers. On a spring day they cycle down straight roads with