 |
|
 | 1. Explain why there is a significant difference between
Figures 0.35a and 0.35b. |
| | 2. In
Figure 0.35a, the graph approaches the lower portion of the
vertical asymptote from the left, whereas the graph approaches the upper portion of the
vertical asymptote from the right. Use the table of function values below the fig-ure to explain how to determine whether a graph approaches a
vertical asymptote by dropping down or rising up. |
| | 3. In the text, we discussed the difference between graphing with a fixed window versus an automatic window. Discuss the advantages and disadvantages of each. (Hint: Consider the case of a first graph of a
function you know nothing about and the case of hoping to see the important details of a graph for which you know the general shape.) |
| | 4. Examine the graph of  with each of the following graphing windows: (a) - 10 x 10 , (b)
. Explain why the graph in (b) doesn't show the details that graph (a) does. |
| In exercises 5-34, sketch a graph of the
function showing all extrema, intercepts, and asymptotes. |
| 5. f (x) = x2 - 1 | 6. f (x) = 3 - x2 |
| 7. f (x) = x2 + 2x + 8 | 8. f (x) = x2 - 20x + 11 |
| 9. f (x) = x3 + 1 | 10. f (x) = 10 - x3 |
| 11. f (x) = x3 + 2x - 1 | 12. f (x) = x3 - 3x + 1 |
| 13. f (x) = x4 - 1 | 14. f (x) = 2 - x4 |
| 15. f (x) = x4 + 2x - 1 | 16. f (x) = x4 - 6x2 + 3 |
| 17. f (x) = x5 + 2 | 18. f (x) = 12 - x5
|
| 19. f (x) = x5 - 8x3 + 20x - 1
|
| 20. f (x) = x5 + 5x4 + 2x3 + 1
|
| 21.  | 22.  |
| 23.  | 24.  |
| 25.  | 26. 
|
| 27.  | 28.  |
| 29.  | 30.  |
| 31.  | 32.  |
| 33.  | 34. 
|
| In exercises 35-44, use a graph to find all
intercepts.
|
| 35. f (x) = x2 - 4 | 36. f (x) = x2 - 9 |
| 37. f (x) = x2 + 4 | 38. f (x) = x2 + 9 |
| 39. f (x) = x2 - 3x - 4 | 40. f (x) = x2 + 6x + 8
|
| 41. f (x) = x3 - 2x2 - x + 2
| 42. f (x) = x3 + 2x2 - x - 2
|
| 43.  | 44. 
|
| In exercises 45-54, find all vertical asymptotes.
|
| 45.  | 46.  |
| 47.  | 48.  |
| 49.  | 50.  |
| 51.  | 52.  |
| 53.  | 54. 
|
| In exercises 55-58, a standard graphing window will not reveal all of the important details of the graph. Adjust the graphing window to find the missing details.
|
| 55. f (x) =  x3 -  x
|
| 56. f (x) = x4 - 11x3 + 5x - 2
|
| 57. 
|
| 58. f (x) =  x5 -  x4 + x3 +  x2 - 6x
|
| In exercises 59-64, adjust the graphing window to identify all horizontal asymptotes.
|
| 59.  | 60.  |
| 61.  | 62.  |
| 63.  | 64. 
|
| 65. Graph y = x2 with the graphing window - 10 x 10 , - 10 y 10 , without drawing the x - and y - axes. Adjust the graphing window for y = 2(x -1)2 + 3 so that (without the axes showing) the graph looks identical to that of y = x2. |
| 66. Graph y = x2 with the graphing window - 10 x 10 , - 10 y 10. Separately graph y = x4 with the same graphing window. Compare and contrast the graphs. Then graph the two functions on the same axes and carefully examine the differences with -1 < x < 1 and x > 1. |
| 67. In this exercise, you will find an equation describing all points equidistant from the x - axis and the point (0, 2). First, see if you can sketch a picture of what the curve through these points ought to look like. For a point (x, y) that is on the curve, explain why  . Square both sides of this equation and solve for y. Identify the curve. |
| 68. Find an equation describing all points equidistant from the x-axis and (1,4) (see exercise 67). |
| 69. A falling object will approach a
constant speed known as the terminal velocity, denoted vT. For a sphere with weight W and diameter D,  for some
constant c. According to legend, Galileo dropped a wood ball and a lead ball of the same size from the Leaning Tower of Pisa to prove that the
acceleration due to gravity does not depend on weight. (This is true, but the effect of air resistance does depend on weight.) If the lead ball is twice as heavy as the wood ball, how will the terminal velocities compare? |
| 70. Fill in the blank: a ball that falls four times as fast as a wood ball is times as heavy (see exercise 69). |
| 71. Suppose a graphing calculator is set up with pixels corresponding to x = 0,0.1, 0.2, 0.3, , 2.0 and y = 0,0.1,0.2,0.3,,4.0. For the function f (x) = x2 , compute the indicated
function values and round off to give pixel coordinates [e.g., the point (1.19,1.4161) has pixel coordinates (1.2,1.4) ]. (a) f (0.4) , (b) f (0.39) , (c) f (1.17) , (d) f (1.20) , (e) f (1.8) , (f) f (1.81). Repeat (c) - (d) if the graphing window is zoomed in so that x = 1.00, 1.01, , 1.20 and y = 1.30, 1.31, , 1.50. Repeat (e) - (f) if the graphing window is zoomed in so that x = 1.800, 1.801, , 1.820 and y = 3.200 3.205,,3.300. |
|
| 72. Graph y = x2 - 1 , y = x2 + x - 1 , y = x2 + 2x -1, and y = x2 - 2x - 1 and other functions of the form y = x2 + cx - 1. Describe the effect(s) a change in c has on the graph. |
|
| 73. Figures 0.31 and
0.32 provide a catalog of the possible types of graphs of
cubic polynomials. In this exercise, you will compile a catalog of graphs of fourth-order polynomials (i.e., y = ax4 + bx3 + ). Start by using your calculator or computer to sketch graphs with different values of a, b, c, d and e. Try y = x4 , y = 2x4 , y = - 2x4 , y = x4 + x3 , y = x4 + 2x3 , y = x4 - 2x3 , y = x4 + x2 , y = x4 - x2 , y = x4 - 2x2 , y = x4 + x , y = x4 - x and so on. Try to determine what effect each
constant has. |
|
| 74. When tracking the motion of an object such as a comet, it is often convenient to think of the object's position as a
function of time. In two dimensions, the object's position would be given by functions x(t ) and y(t ). This is an example of parametric equations. The relatively simple
polynomial graphs of this section can interact to produce complicated parametric graphs. Use a graphics calculator or computer software that graphs parametric equations to graph x(t ) = t 2 - 2 and y(t ) = t 3 - t - 1. Find the point (x, y) (and the corresponding values of t ) where the loop begins. Try to explain in terms of the properties of x(t ) and y(t ) why it loops. |
