Warm-Up 1.5 –2 Evaluate the expression without using a calculator. ANSWER –24 4. State the domain and range of the function y = –(x – 2) domain: all real numbers; range: y ≤ 3 3.–

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. Cycle# of m & m’s

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. Cycle# of m & m’s

Exponential Growth Functions 4.1 (M3) P. 130

Vocabulary Exponential function: y = ab x (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

EXAMPLE 1 Graph y = b for b > 1 x SOLUTION Make a table of values.STEP 1 STEP 2 Plot the points from the table. Graph y =. x 2 STEP 3 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.

EXAMPLE 2 Graph y = ab for b > 1 x Graph the function. a. y = x SOLUTION 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. 0, 1 2 a.

EXAMPLE 2 Graph the function. Graph y = ab for b > 1 x b. y = – 5 2 x 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. b.b. 1, – 5 2 SOLUTION

EXAMPLE 3 Graph y = ab + k for b > 1 x–hx–h Graph y = 4 2 – 3. State the domain and range. x – 1 SOLUTION 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 – 1

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

EXAMPLE 4 Solve a multi-step problem Write an exponential growth model giving the number n of incidents t years after About how many incidents were there in 2003 ? 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

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 = So, the exponential growth model is: n = a (1 + r) t = 2573( ) t = 2573(1.92) t Write exponential growth model. Substitute 2573 for a and 0.92 for r. Simplify.

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

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

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

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

EXAMPLE 5 Find the balance in an account 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. FINANCE a. Quarterly b. Daily

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

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

GUIDED PRACTICE for Example 5 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. $ ANSWER a. With interest compounded daily, the balance after 3 years is: