Warm-Up

1. 3. –3 4 1 ANSWER 2 3 25 2 3 2. 8 4 ANSWER Warm-Up Evaluate the expression without using a calculator. 1. 5–2 –24 4.State the domain and range of the function y = –(x – 2)2 + 3. domain: all real numbers; range: y ≤ 3

2. Exponential Graphs with M & M’s!!! • Make a t chart as shown • Start with 1 m & m • For each cycle, double the number of m & m’s you have on your paper towel (record the number each time) • Continue until you finish the chart • Plot the points on your graphing calculator • Look at the graph and use regression to make the equation.

3. Now let’s do Exponential Decay! • Create another t chart this time starting with 32 m & m’s. • ½ the m & m’s each time (You may eat them when you record your number.) • Continue the chart. • Plot the points and use the regression key to come up with the equation.

4. Exponential Growth Functions 4.1 (M3) P. 130

5. Vocabulary • Exponential function: y = abx (x is the exponent) • If a>0 and b>1, then it is exponential growth. • B is growth factor • Asymptote: line a graph approaches but never touches • basic exponential graphs have 1 asymptote • Exponential Growth Model y = a(1+r)t, where t is time, a is initial amount and r is the % increase • 1 + r is the growth factor

6. x 2 Graph y = . x EXAMPLE 1 Graph y=b for b > 1 SOLUTION STEP 1 Make a table of values. STEP 2 Plot the points from the table. Draw, from left to right, a smooth curve that begins just above the x-axis, passes through the plotted points, and moves up to the right. STEP 3

7. x 1 1 Graph y = ab for b > 1 a. Plot and (1, 2) .Then, from left to right, draw a curve that begins just above the x-axis, passes through the two points, and moves up to the right. 2 2 0, EXAMPLE 2 Graph the function. x a.y = 4 SOLUTION

8. 1,– x b. y = – x 5 5 Graph y = ab for b > 1 b. 2 2 Plot (0, –1) and . Then,from left to right, draw a curve that begins just below the x-axis, passes through the two points,and moves down to the right. EXAMPLE 2 Graph the function. SOLUTION

9. x– 1 Graphy = 4 2 – 3.State the domain and range. Begin by sketching the graph of y = 4 2 , which passes through (0, 4) and (1, 8). Then translate the graph right 1 unit and down 3 units to obtain the graph of y = 4 2 – 3.The graph’s asymptote is the line y = –3. The domain is all real numbers, and the range is y > –3. x x–h Graph y = ab + k for b > 1 x– 1 EXAMPLE 3 SOLUTION

10. x 1. y = 4 x 2. y = 3 x + 1 2 3. f (x) = 3 + 2 3 for Examples 1, 2 and 3 GUIDED PRACTICE Graph the function. State the domain and range.

11. EXAMPLE 4 Solve a multi-step problem In 1996, there were 2573 computer viruses and other computer security incidents. During the next 7 years, the number of incidents increased by about 92% each year. Computers • Write an exponential growth model giving the number n of incidents tyears after 1996. About how many incidents were there in 2003?

12. t t = 2573(1.92) n = a(1 + r) t = 2573(1 + 0.92) EXAMPLE 4 Solve a multi-step problem • Graph the model. • Use the graph to estimate the year when there were about 125,000 computer security incidents. SOLUTION STEP 1 The initial amount is a = 2573 and the percent increase is r =0.92. So, the exponential growth model is: Write exponential growth model. Substitute 2573 for aand 0.92 for r. Simplify.

13. Using this model, you can estimate the number of incidents in 2003 (t = 7) to be n = 2573(1.92) 247,485. 7 EXAMPLE 4 Solve a multi-step problem STEP 2 The graph passes through the points (0, 2573) and (1,4940.16). Plot a few other points. Then draw a smooth curve through the points.

14. Using the graph, you can estimate that the number of incidents was about 125,000 during 2002 (t 6). EXAMPLE 4 Solve a multi-step problem STEP 3

15. for Example 4 GUIDED PRACTICE 4.What If?In Example 4, estimate the year in which there were about 250,000 computer security incidents. SOLUTION 2003

16. x 5.In the exponential growth model y = 527(1.39) , identify the initial amount,the growth factor, and the percent increase. for Example 4 GUIDED PRACTICE SOLUTION Initial amount: 527Growth factor 1.39Percent increase 39%

17. FINANCE You deposit $4000 in an account that pays 2.92% annual interest. Find the balance after 1 year if the interest is compounded with the given frequency. EXAMPLE 5 Find the balance in an account a.Quarterly b.Daily

18. = 4000 1 + 0.0292 r 4 n 4 1 4 = 4000(1.0073) nt A = P 1 + EXAMPLE 5 Find the balance in an account SOLUTION a.With interest compounded quarterly, the balance after 1 year is: Write compound interest formula. P = 4000, r = 0.0292, n = 4, t = 1 Simplify. Use a calculator. = 4118.09 ANSWER The balance at the end of 1 year is $4118.09.

19. 365 1 = 4000 1 + r 0.0292 365 n 365 = 4000(1.00008) ANSWER nt A = P 1 + The balance at the end of 1 year is $4118.52. EXAMPLE 5 Find the balance in an account b. With interest compounded daily, the balance after 1 year is: Write compound interest formula. P = 4000, r = 0.0292, n = 365, t = 1 Simplify. Use a calculator. = 4118.52

20. ANSWER $2254.98 for Example 5 GUIDED PRACTICE 6. FINANCE You deposit $2000 in an account that pays 4% annual interest. Find the balance after 3 years if the interest is compounded daily. a.With interest compounded daily, the balance after 3 years is: