Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
Подождите немного. Документ загружается.
EXAMPLE 7
Sketch a complete graph of
y .007x
5
.2x
4
1.332x
3
.004x
2
10.
SOLUTION Four different viewing windows for this graph are shown in
Figure 2–19. Graph (a) (the standard window) is certainly not complete, since it
shows no points to the right of the y-axis. Graph (b) is not complete, since it indi-
cates that parts of the graph lie outside the window.
86 CHAPTER 2 Graphs and Technology
10
−10
(a)
−10 10
40
−40
(b)
−20 20
120
−320
(c)
−10 20
15,000
−15,000
(d)
−100 100
Figure 2–19
Graph (d) tends to confirm what Graph (c) suggests: that the graph keeps climb-
ing sharply forever as you move to the right and that it keeps falling sharply
forever as you move to the left. Because of its large scale, however, graph (d)
doesn’t show the features of the graph near the origin. So we conclude that graph (c)
is probably a complete graph, since it shows important features (twists and turns)
near the origin, as well as suggesting the shape of the graph farther out. ■
EXAMPLE 8
If you graph y 2x
3
26x
2
18x 50 in the standard viewing window,
you get a blank screen (try it!). In such cases, you can usually find at least one
point on the graph by setting x 0 and determining the corresponding value of y
(the y-intercept of the graph). If x 0 here, then y 50, so the point (0, 50) is on
TECHNOLOGY TIP
Most calculators have an “auto scal-
ing” feature. Once the range of x
values has been set, the calculator
selects a viewing window that includes
all the points on the graph whose
x-coordinates are in the chosen range.
It is in the ZOOM menu (VIEWS
menu on HP-39gs) and is called
ZoomFit, ZFit, or Auto(Scale).
This feature can eliminate some
guesswork but may produce a window
so large that it hides some of the
features of the graph.
the graph. Consequently, the y-axis of our viewing window should extend well
beyond 50. An alternative method of finding some points on the graph is in the
Technology Tip in the margin.
SCATTER PLOTS AND LINE GRAPHS
A somewhat different procedure must be used to graph scatter plots and construct
line graphs on a calculator.
EXAMPLE 9
The winning speeds for the Indianapolis 500 race (rounded to the nearest mph) are
shown in the table.
SECTION 2.1 Graphs 87
Year 2000 2001 2002 2003 2004 2005 2006
Speed 168 142 166 156 139 158 157
For this data, construct a
(a) scatter plot
(b) line graph.
SOLUTION Consult the Technology Tip after the example for directions on
how to carry out the steps listed below.
(a) Let x 0 correspond to 2000, so that the data points are (0, 168), (1, 142),
(2, 166), and so on. Enter these points in the statistics editor as two lists
(x-coordinates in the first list and the corresponding y-coordinates in the
second), as shown in Figure 2–20. In the stat plot setup screen (Figure 2–21),
select “scatter plot” as the type. Enter an appropriate viewing window and
press GRAPH to obtain the scatter plot in Figure 2–22.
1
0
200
7
Figure 2–21
Figure 2–20
Figure 2–22
Find a complete graph of this equation. [Hint: The graph crosses the x-axis once and
has one “peak” and one “valley.”]
■
GRAPHING EXPLORATION
TECHNOLOGY TIP
When you get a blank screen, press
TRACE and use the left/right arrow
keys. The coordinates of points on the
graph will be displayed at the bottom
of the screen, even though they aren’t
in the current viewing window. Use
these coordinates as a guide for
selecting a viewing window in which
the graph does appear.
88 CHAPTER 2 Graphs and Technology
1
0
200
7
Figure 2–23
*When statistical computations are run on TI-86, the lists used are automatically copied into the x-stat
and y-stat lists, replacing whatever was there before. So use the x-stat and y-stat lists only if you don’t
want to save them. See your instruction manual to find out how to create new lists in the statistics
editor.
†
Types are shown schematically on TI-84. The shaded one in Figure 2–21 indicates a scatter plot and
the one to its right a line graph. Scatter plots and line graphs are listed as “scat” and “xy line”
respectively on TI-86 and Casio 9850. On HP-39gs, check “connect” on the second page of the stat
plot setup screen for a line graph and uncheck it for a scatter plot. To enter the lists to be graphed on
HP, press SYMB (ignore the “fit” setting).
‡
On Casio, the viewing window will be chosen automatically unless “stat wind” in the main setup
menu (SHIFT MENU) is set to “manual” (recommended). On HP, the viewing window is selected
in the plot setup screen. If a line or curve appears on the scatter plot, press MENU and FIT to
remove it.
(b) For a line graph of the data, change the type to “line graph” in the stat plot
setup screen. Then press GRAPH to obtain Figure 2–23. ■
TECHNOLOGY TIP
Most calculators allow you to store three or more statistics graphs (identified by number); the
following directions assume that the first one is used.
To call up the statistics editor, use these commands:
TI-84: STAT EDIT (lists are L
1
, L
2
, . . .);
TI-86: STAT EDIT (built-in lists are x-stat, y-stat; it is usually better to create your own
lists; we use L1 and L2 as list names here);*
TI-89: APPS DATA/MATRIX EDITOR NEW; then choose DATA as the TYPE, enter a VARIABLE
name (we use L here), and key in ENTER (lists are C
1
, C
2
, . . .);
Casio 9850: STAT (lists are List 1, List 2, . . .);
HP-39gs: APLET STATISTICS (lists are C
1
, C
2
, . . .).
Enter the data points as two lists (x-coordinates in the first, and the corresponding
y-coordinates in the second).
To create a scatter plot or line graph of the data points (after the lists have been
entered), call up the stat plot setup screen with these commands:
TI-84: STAT PLOT (on keyboard) PLOT 1;
TI-86: PLOT (on STAT menu) PLOT 1;
TI-89: PLOT-SETUP and DEFINE (in the Data Editor);
Casio 9850: GRPH SET (on the STAT list screen);
HP-39gs: PLOT-SETUP (on the keyboard)
Enter the request information, such as graph number, lists to be used, the mark used for the
data points, and the plot type.
†
Finally, select a viewing window and press GRAPH on TI (or
GPH 1 on Casio or PLOT on HP).
‡
SECTION 2.1 Graphs 89
In Exercises 1–6, graph the equation by hand by plotting no
more than six points and filling in the rest of the graph as best
you can. Then use the calculator to graph the equation and
compare the results.
1. y x 2 2. y x 5
3. y x
2
x 4. y x
2
x 1
5. y x
3
1 6. y
1
x
In Exercises 7–12, find the graph of the equation in the stan-
dard window.
7. 3 y .5x 8. y 2x 4
9. y x
2
5x 2 10. y .3x
2
x 4
11. y .2x
3
.1x
2
4x 1
12. y .2x
4
.2x
3
2x
2
2x 5
13. (a) Graph
y
x
2
1
1
in the standard window.
(b) Does the graph appear to stop abruptly partway along
the x-axis? Use the trace feature to explain why this
happens. [Hint: In this viewing window, each pixel
represents a rectangle that is approximately .32 unit
high.]
(c) Find a viewing window with 10 x 10 that shows
a complete graph that does not fade into the
x-axis.
14. (a) Graph y x
3
2x
2
x 2 in the standard window.
(b) Use the trace feature to show that the portion of the
graph with 0 x 1.5 is not actually horizontal. [Hint:
All the points on a horizontal segment must have the
same y-coordinate (why?).]
(c) Find a viewing window that clearly shows that the
graph is not horizontal when 0 x 1.5.
In Exercises 15–24, use the techniques of Examples 4 and 5 to
graph the equation in a suitable square viewing window.
15. x
2
y
2
16 16. y
2
x 2
17. 3x
2
2y
2
48
18. 25(x 5)
2
36(y 4)
2
900
19. (x 4)
2
(y 2)
2
25
20. 9x
2
4y
2
36 21. 4x
2
9y
2
36
22. 9y
2
x
2
9 23. 9x
2
5y
2
45
24. x y
2
2
25. Use your minimum finder to approximate the x-coordinates
of the lowest point on the graph of y x
3
2x 5 in the
window with 0 x 5 and 3 y 8. The correct
answer is
x
2
3
.816496580928.
How good is your approximation?
In Exercises 26 and 27, use zoom-in or a maximum/minimum
finder to determine the highest and lowest point on the graph
in the given window.
26. y .07x
5
.3x
3
1.5x
2
2(3 x 2
and 6 y 6)
27. y .04x
4
.01x
3
1.7x
2
2.5x 3(0 x 5
and 5 y 5)
28. According to United Nations projections, the population of
China (in millions) between now and 2050 is given by
y .00096x
3
.1x
2
11.3x 1274 (0 x 50),
where x 0 corresponds to 2000. In what year will China
reach its maximum population, and what will that popula-
tion be?
29. The national average interest rate for a 30-year fixed rate
mortgage is approximated by
y .015x
3
.112x
2
1.562x 8.67 (0 x 6),
where x 0 corresponds to 2000.* According to this model,
when was the rate the lowest? What was the lowest rate?
30. The following equation gives the approximate number of
thefts at Cleveland businesses each hour of the day:
y .00357x
4
.3135x
3
6.87x
2
38.3x 118.4
(0 x 23),
where x is measured in hours after midnight.
†
(a) About how many thefts occurred around 5 A.M.?
(b) When did the largest number of thefts occur?
In Exercises 31–36, determine which of the following viewing
windows gives the best view of the graph of the given equation.
(a) 10 x 10; 10 y 10
(b) 5 x 25; 0 y 20
(c) 10 x 10; 100 y 100
(d) 20 x 15; 60 y 250
(e) None of a, b, c, d gives a complete graph.
31. y 18x 3x
2
32. y 4x
2
80x 350
33. y
1
3
x
3
25x 100 34. y x
4
x 5
35. y x
2
50x 625 36. y .01(x 15)
4
*http://mortgage-x.com
†
Cleveland Police Department
EXERCISES 2.1
In Exercises 37–42, obtain a complete graph of the equation by
trying various viewing windows. List a viewing window that
produces this complete graph. (Many correct answers are pos-
sible; consider your answer to be correct if your window shows
all the features in the window given in the answer section.)
37. y 7x
3
35x 10 38. y x
3
5x
2
5x 6
39. y x
2
x 40. y 1/x
2
41. y .1x
4
x
3
x
2
x 50
42. y .002x
5
.06x
4
.001x
3
.04x
2
.2x 15
In Exercises 43–46, use technology to construct a scatter plot
and a line graph of the data.
43. Last year’s electric bills for one of the authors are shown in
the table. Let x 3 correspond to March, x 4 to April, etc.
44. The table shows the population of Kansas City, Missouri in
various years.* Let x 0 correspond to 1950.
45. The table gives the average SAT math score in 2005 for stu-
dents in states with a population of 5 to 6 million people.
†
Let x 1 be the first state on the list, x 2 the second, etc.
90 CHAPTER 2 Graphs and Technology
46. The average monthly rainfall (in inches) in Cleveland, Ohio
(based on a thirty year average) is shown in the table.* Let
x 1 correspond to January, x 3 to March, etc.
47. (a) Graph y .3(x 3)
4
8 in the standard window.
(b) To see exactly which points the calculator actually
graphed, change the graphing mode to Dot (or Draw
Dot or Plot) in the menu/submenu list below, and graph
again.
TI-84: MODE
TI-86: GRAPH/FORMAT
TI-89: Y/STYLE
Casio: SETUP/DRAW TYPE
HP-39gs: uncheck “connect” on the second page of
the PLOT SETUP menu.
(c) Why does the graph in part (b) look “solid” at the top,
but consists of isolated points elsewhere?
48. (a) Graph y .3x
3
2x
2
6 in the standard window.
(b) Use trace to move to a point whose x-coordinate is close
to 1.
(c) Set the zoom factors of your calculator to 10. Zoom-in
once or twice. Does the graph appear to be a straight
line near the point?
(d) Repeat parts (a)—(c) at lowest point to the right of the
y-axis. Is the result the same? If not, keep zooming in
until it is (at each stage move the flashing cursor up or
down, so it is on the graph).
(e) What do parts (a)—(d) suggest about the graph?
In Exercises 49–54, use your algebraic knowledge to state
whether or not the two equations have the same graph. Con-
firm your answer by graphing the equations in the standard
window.
49. y x 3 and y x 3
50. y x 4 and y x 4
51. y x
2
and y x
52. y x
2
6
x 9
and y x 3
53. y x
2
9
and y x 3
Month Bill ($)
March 61
April 50
May 116
June 187
July 149
August 182
September 77
Year 1950 1960 1970 1980 1990 2004
Population 457 476 507 448 435 444
*U.S. Census Bureau
†
The College Board
State Score
Arizona 530
Indiana 508
Maryland 515
Minnesota 597
Missouri 588
Tennessee 563
Washington 534
Wisconsin 599
*National Climatic Data Center
Month Rainfall
January 2.5
March 2.9
May 3.5
July 3.5
September 3.8
November 3.4
54. y
x
2
1
2
and y
x
1
2
1
2
55. (a) Confirm the accuracy of the factorization
x
2
5x 6 (x 2)(x 3) graphically.
[Hint: Graph y x
2
5x 6 and y (x 2)(x 3) on
the same screen. If the factorization is correct, the
graphs will be identical (which means that you will see
only a single graph on the screen).]
(b) Show graphically that (x 5)
2
x
2
5
2
. [Hint: Graph
y (x 5)
2
and y x
2
5
2
on the same screen. If the
graphs are different, then the two expressions cannot be
equal.]
True or False. In Exercises 56–58, use the technique of
Exercise 55 to determine graphically whether the given state-
ment is possibly true or definitely false. (We say “possibly
true” because two graphs that appear identical on a calculator
screen may actually differ by small amounts or at places not
shown in the window.)
56. x
3
7x 6 (x 1)(x 2)(x 3)
57. (1 x)
6
1 6x 15x
2
20x
3
15x
4
6x
5
x
6
58. x
5
8x
4
16x
3
5x
2
4x 20
(x 2)
2
(x 5)(x
2
x 1)
59. A toy rocket is shot straight up from ground level and then
falls back to earth; wind resistance is negligible. Use your
calculator to determine which of the following equations
has a graph whose portion above the x-axis provides the
most plausible model of the path of the rocket.
(a) y .1(x 3)
3
.1x
2
5
(b) y x
4
16x
3
88x
2
192x
(c) y 16x
2
117x
(d) y .16x
2
3.2x 16
(e) y (.1x 3)
6
600
60. Monthly profits at DayGlo Tee Shirt Company appear to be
given by the equation
y .00027(x 15,000)
2
60,000,
where x is the number of shirts sold that month and y is
the profit. DayGlo’s maximum production capacity is
15,000 shirts per month.
(a) If you plan to graph the profit equation, what range of x
values should you use? [Hint: You can’t make a nega-
tive number of shirts.]
(b) The president of DayGlo wants to motivate the sales
force (who are all in the profit-sharing plan), so he asks
you to prepare a graph that shows DayGlo’s profits in-
creasing dramatically as sales increase. Using the profit
equation and the x range from part (a), what viewing
window would be suitable?
(c) The City Council is talking about imposing more taxes.
The president asks you to prepare a graph showing that
DayGlo’s profits are essentially flat. Using the profit
equation and the x range from part (a), what viewing
window would be suitable?
SECTION 2.1 Graphs 91
In each of the applied situations in Exercises 61–64, find an
appropriate viewing window for the equation (that is, a win-
dow that includes all the points relevant to the problem but
does not include large regions that are not relevant to the prob-
lem, and has easily readable tick marks on the axes). Explain
why you chose this window. See the Hint in Exercise 60(a).
61. Beginning in 1905 the deer population in a region of
Arizona rapidly increased because of a lack of natural
predators. Eventually food resources were depleted to such
a degree that the deer population completely died out. In
the equation y .125x
5
3.125x
4
4000, y is the num-
ber of deer in year x, where x 0 corresponds to 1905.
62. A cardiac test measures the concentration y of a dye x sec-
onds after a known amount is injected into a vein near the
heart. In a normal heart
y .006x
4
.14x
3
.053x
2
179x.
63. The concentration of a certain medication in the blood-
stream at time x hours is approximated by the equation
y
.1x
3
3
7
5x
50
,
where y is measured in milligrams per liter. After two days
the medication has no effect.
64. A winery can produce x barrels of red wine and y barrels of
white wine, where
y
200
2
,
0
0
0
0
0
0
x
50x
.
In Exercises 65–67, use the viewing windows found in
Exercises 61–63.
65. (a) Use the trace feature to estimate the year when the deer
population in Exercise 61 first reached 40,000. In what
later year was it also 40,000?
(b) When was the deer population at its maximum?
Approximately how many deer were there at that time?
66. (a) In the cardiac test of Exercise 62, use the trace feature to
estimate the time at which all dye is gone from the body.
(b) At what time during the test was the concentration of
the dye the greatest?
67. (a) When was the concentration of the medication in Exer-
cise 63 at its peak?
(b) Use the trace feature to determine approximately when
the concentration dropped below 10 milligrams per liter
and stayed below that level.
68. The total resources (in billions of dollars) of the Pension
Benefit Guaranty Corporation, the government agency that
insures pensions, is approximated by
y .279x
2
4.006x 28.412 (4 x 20),
where x 4 corresponds to 2004.*
*Center on Federal Financial Institutions
(continued)
(a) When are resources the greatest?
(b) Use the trace feature to find the approximate time when
the Corporation will run out of money.
In Exercises 69–72, graph all four equations on the same
screen, using a sufficiently large square viewing window, and
answer this question: What is the geometric relationship of
graphs (b), (c), and (d) to graph (a)?
69. (a) y x
2
(b) y x
2
5
(c) y x
2
5 (d) y x
2
2
70. (a) y x
(b) y x 3
(c) y x 3
(d) y x 6
71. (a) y x
(b) y 2x
(c) y 3 x
(d) y
1
2
x
72. (a) y x
2
(b) y x
2
(c) y
1
2
x
2
(d) y 2x
2
In Exercises 73–75, graph the two given equations and the
equation y x on the same screen, using a sufficiently large
square viewing window, and answer this question: What is the
geometric relationship between graphs (a) and (b)?
73. (a) y x
3
(b) y
3
x
92 CHAPTER 2 Graphs and Technology
74. (a) y
1
2
x
3
4 (b) y
3
2x 8
75. (a) y 5x 15 (b) y .2x 3
76. Put your calculator in radian mode, and use the viewing
window given by 0 x 6.28 and 2 y 2.*
(a) Graph y sin x
(b) Graph y sin (2x)
(c) Graph y sin (3x)
(d) On the basis of parts (a)–(c), what do you think the
graphs of y sin (4x), y sin (5x), y sin (6x), and so
on, will look like? Use the calculator to verify your
answer.
(e) On the basis of part (d), what do you think the graphs of
y sin (50x) and y sin (100x) will look like? What
does a calculator display instead? What might explain
the graphs of the calculator?
*You don’t need to know what radian mode is or what “sin” means. Just
use the calculator key with this label.
2.2 Solving Equations Graphically and Numerically
■ Use technology to solve equations graphically and numerically.
■ Solve equations by the intercept method.
■ Solve equations the intersection method.
■ Use an equation solver to solve equations numerically.
■ Learn strategies for solving radical and fractional equations with
technology.
Algebraic techniques can be used to solve linear, quadratic, and some higher-
degree equations, as we saw in Section 1.2. For many other equations, however,
graphical and numerical approximation methods are the only practical alternatives.
These methods are based on a connection between equations and graphs that we
now examine.
Recall that the x-intercepts of the graph of
y x
4
4x
3
3x
2
x 2
are the x-coordinates of the points where the graph intersects the x-axis (see Fig-
ure 2–24). Since points on the x-axis have 0 second coordinates, the x-intercepts are
found algebraically by setting y 0 and solving for x, that is, by solving the equation
x
4
4x
3
3x
2
x 2 0.
Graphical equation solving amounts to running this in the opposite direction: An
equation is solved by finding the x-intercepts, as illustrated in the next example.
Section Objectives
10
−10
x-intercepts
−66
Figure 2–24
EXAMPLE 1
Solve x
4
4x
3
3x
2
x 2 0 graphically.
SOLUTION As we have just seen, the solutions are the x-intercepts of the
graph of
y x
4
4x
3
3x
2
x 2.
These x-intercepts may be found by graphing the equation and using the graphi-
cal root finder, as in Figure 2–25. (See the Technology Tip in the margin.)
SECTION 2.2 Solving Equations Graphically and Numerically 93
10
−10
(a)
−66
10
−10
(b)
−66
Figure 2–25
Therefore, the approximate solutions of the original equation are x .5242 and
x 3.2257. ■
The process in Example 1 works in the general case as well.
The Intercept
Method
To solve a one-variable equation of the form,
Expression in x 0,
1. Graph the two-variable equation
y expression in x.
2. Use a graphical root finder to determine the x-intercepts of the graph.
The x-intercepts are the real solutions of the equation.
EXAMPLE 2
Solve x
3
4x
2
2x 5 0.
SOLUTION We graph y x
3
4x
2
2x 5 (Figure 2–26) and see that it has
three x-intercepts. So the equation has three real solutions. A root finder shows
that one solution is x 1.1926 (Figure 2–27 on the next page). ■
TECHNOLOGY TIP
The graphical root finder is labeled
Root or Zero and is in this menu/
submenu:
TI-84: CALC
TI-86/89: GRAPH/MATH
Casio: DRAW/G-SOLV
HP-39gs: PLOT/FCN
On most TI calculators, you must se-
lect a left and a right bound (x-values
to the left and right of the intercept)
and make an initial guess.
10
−10
−10 10
Figure 2–26
94 CHAPTER 2 Graphs and Technology
10
−10
−10 10
Figure 2–27
THE INTERSECTION METHOD
The next examples illustrate an alternative graphical method, which is sometimes
more convenient for solving equations.
EXAMPLE 3
Solve x
2
4x 3 x
3
x 6.
SOLUTION Let y
1
x
2
4x 3 and y
2
x
3
x 6, and graph both
equations on the same screen (Figure 2–28). Consider the point where the
two graphs intersect. Since it is on the graph of y
1
, its second coordinate is
x
2
4x 3, and since it is also on the graph of y
2
, its second coordinate is
x
3
x 6. So for this number x, we must have x
2
4x 3 x
3
x 6.
In other words, the x-coordinate of the intersection point is the solution of the
equation.
10
−10
−10 10
Figure 2–28
10
−10
−10 10
Figure 2–29
This coordinate can be approximated by using a graphical intersection finder (see
the Technology Tip in the margin), as shown in Figure 2–29. Therefore, the solu-
tion of the original equation is x 2.207. ■
TECHNOLOGY TIP
The graphical intersection finder is in
the same menu/submenu as the root
finder. It is labeled
Intersection,
Intersect, Isect, or ISCT.
Use your graphical root finder to find the other two solutions (x-intercepts) of the
equation in Example 2.
GRAPHING EXPLORATION
The technique used in Example 3 and the preceding exploration works in the
general case
SECTION 2.2 Solving Equations Graphically and Numerically 95
The Intersection
Method
To solve an equation of the form
First expression in x Second expression in x,
1. Set y
1
equal to the left side and y
2
equal to the right side.
2. Graph y
1
and y
2
on the same screen.
3. Use a graphical intersection finder to find the x-coordinate of each point
where the graphs intersect.
These x-coordinates are the real solutions of the equation.
EXAMPLE 4
According to data from the U.S. Census Bureau, the approximate population y (in
millions) of Detroit is given by
y (6.985 10
7
)x
4
(9.169 10
5
)x
3
.00373x
2
.0268x 1.85,
where x 0 corresponds to 1950. The approximate population of San Diego dur-
ing the same period is given by
y (6.216 10
6
)x
3
(4.569 10
4
)x
2
.01035x .346.
(a) Approximately when did the two cities have the same population?
(b) In what year was the population of Detroit about 1,500,000?
SOLUTION
(a) The populations of the two cities are equal when
(6.985 10
7
)x
4
(9.169 10
5
)x
3
.00373x
2
.0268x 1.85
(6.216 10
6
)x
3
(4.569 10
4
)x
2
.01035x .346.
We solve this equation by graphing the left side as y
1
and the right side as y
2
and
finding the intersection point, as in Figure 2–30. Rounding the x-coordinate to
the nearest year (x 38), we see that the populations were equal around 1988.
(b) To find when the population of Detroit was 1,500,000 (that is, 1.5 million),
we must solve the equation
(6.985 10
7
)x
4
(9.169 10
5
)x
3
.00373x
2
.0268x 1.85 1.5.
Since the left side of this equation has already been graphed as y
1
, we need
only graph y
3
1.5 on the same screen and find the intersection point of y
1
and y
3
, as in Figure 2–31. The solution x 20.45 shows that Detroit had a
population of 1.5 million in 1970. ■
3
0
60
Figure 2–30
Show that the equation x
2
2x 6 2x 7
has two real solutions by graphing
the left and right sides in the standard window and counting the number of intersec-
tion points. The positive solution is x 4.3094. Find the negative solution.
GRAPHING EXPLORATION
3
0
60
Figure 2–31