Use a graphing calculator to work Exercise. (See Examples 1 and 2.)Health The number of do...

Use a graphing calculator to work Exercise. (See Examples 1 and 2.)

Health The number of doctors per 1000 residents in the United States from 1970 to 2020 can be approximated by

where x = 0 corresponds to the year 1970.‡ In what year is the number of doctors the highest? How many doctors are there per 1000 residents in that year?

‡Based on data and projections by Health Affairs and Dr. Richard Cooper, Institute of Helth Policy at the Medical College of Wisconsin.

Example 1

Use a graphical root finder to solve x3 − 5x + 1 = 0.

Solution

First, graph y = x3 − 5x + 1. The x-intercepts of this graph are the solutions of the equation. To find these intercepts, look for “root” or “zero” in the appropriate menu.* Check your instruction manual for the proper syntax. A typical root finder (see Figure 1) shows that two of the solutions (x-intercepts) are x ≈ .2016 and x ≈ 2.1284. For the third solution, see the problem at the side.

*CALC or TI-84+, GRAPH MATH on TI-86, G-SOLV on Casio.

Example 2

Health The percentage y of 12th graders who have ever tried an illicit drug is approximated by

where x = 1 corresponds to the year 1991.* In what year was the percentage of 12th graders who tried illicit drugs the greatest? What was that percentage?

Solution

The graph of y = −.00374x3 − .0591x2 + 2.534x + 38.21 is shown in Figure 2. The highest percentage of 12th graders who tried illicit drugs corresponds to the highest point on this graph. To find this point, look for “maximum” or “max” or “extremum” in the same menu as the graphical root finder. Check your instruction manual for the proper syntax. A typical maximum finder (Figure 3) shows that the highest point has approximate coordinates (10.66, 53.98). The first coordinate is the year and the second is the percentage. So the largest percentage of 12th graders who tried illicit drugs (about 53.98%) occurred in mid-2000 (x ≈ 10.66).

FIGURE 1

FIGURE 2

FIGURE 3

*L. D. Johnston, P. M. O’Malley, J. G. Bachman, and J. E. Schulenberg. “Teen smoking resumes decline.” Ann Arbor. MI: University of Michigan News Service [Online], Available: www.monitoringthefuture.org.