Zero and Negative Eponents Eercises Write each epression as an integer, a simple fraction, or an epression that contains onl positive eponents. Simplif...3 0. 0-0,000 3. a -5. 3.7 0 a 5 5. 9-6. 3-3 9 p 7p 7. (7q) - 8. ( - 7 8) - 7q 6 9 9..8c 0.8 0. (-9.7) 0 Write each epression so that it contains onl positive eponents. Simplif.. -6-3. -rs -5 6 r s 5 3. 7-8 0 7. ( 5a 3b ) - 8 9b 5a w 3 5. (-8v) - w 3 6. 6v -3 m 0 n - n 8 7. (3) 0 z z 8. - 3-3 uv - v 7u
Eponential Functions. For each of the tables on the previous page, etend them two units in each direction. Use the common difference in the -values and the common ratio in the -values to do the etension. The first table is done for ou. 0 3 5 6 7 3 3 9 7 8 3 79 87 656 0.75.5 0 3 6 3 8 5 96 6 9 7 38 0.08 0. 0 0 50 3 50 50 5 650 6 3,50 7 56,50. Plot the points in each of our etended tables on separate coordinate grids. Connect the points with a smooth curve. The domain of each function is all real numbers and that the range is all positive real numbers. Eplain wh there are negative values for but not for. 8000 6000 000 000 O 6 00 300 00 00 O 6 60,000 0,000 80,000 0,000 O 6 The domain is all real numbers because can have an value, but the range is all positive real numbers because a + 0 and b + 0, so a # b will alwas be + 0. 3. For each of the tables, identif the starting value a and the common ratio b. For the first table, a is and b is 3. Net, write the eponential function that describes each table. The function for the first table is f () = # 3. Check if our function is correct b substituting in -values and seeing if the function produces values for that match the values in the table. Table : ; 3; # 3 ; Table : 3; ; 3 # ; Table 3: ; 5; # 5 ;
Comparing Linear and Eponential Functions What is the average rate of change for the function f () = # over the intervals - 0, 0, and? Describe what ou observe. Step Make a table of values. 0 f() 0.5 8 3 6 3 Step Find the average rates of change over the intervals - 0, 0, and. f (b) - f (a) b - a f (b) - f (a) b - a f (b) - f (a) b - a = = = f (0) - f (- ) 0 - (- ) = - 0.5 = 0.75 f () - f (0) - 0 = 8 - = 3 f () - f () - = 3-8 = The average rate of change is different over each interval. It increases with each interval. So the rate of change is increasing. Eercises. Does the table represent a linear function or an eponential function? Eplain. 0 3 5 0 5 30 The difference between each -value is and the difference between each -value is 5. The table represents a linear function because there is a common difference between -values and a common difference between -values.. Suppose that the fish population of a lake decreases b half each week. Can ou model the situation with a linear function or an eponential function? Eplain. The difference between each -value is and the ratio between each -value is 0.5. You can model the situation with an eponential function. 3. Find the average rate of change for the function =.5 # over the intervals 3, 3 5, and 5 7. Describe what ou observe. f(3) f() f(5) f(3) f(7) f(5) = 7.5 = 30 3 5 3 7 5 = 0 The average rate of change is different over each interval. It increases with each interval. So the rate of change is increasing.
Reteaching Eponential Growth and Deca Eponential functions can model the growth or deca of an initial amount. The basic eponential function is = a # b where a represents the initial amount b represents the growth (or deca) factor. The growth factor equals 00% plus the percent rate of change. The deca factor equals 00% minus the percent rate of deca. represents the number of times the growth or deca factor is applied. represents the result of appling the growth or deca factor times. A gm currentl has 000 members. It epects to grow % per ear. How man members will it have in 6 ears? There are 000 members to start, so a = 000. The growth per ear is %, or 0., so b = + 0. or.. The desired time period is 6 ears, so = 6. The function is = 000 #.. When = 6, = 000 #. 6 398. So, the gm will have about 398 members in 6 ears. In Eercises 3, identif a, b, and. Then use them to write the eponential function that models each situation. Finall use the function to answer the question.. When a new bab is born to the Johnsons, the famil decides to invest $5000 in an account that earns 7% interest as a wa to start the bab s college fund. If the do not touch that investment for 8 ears, how much will there be in the college fund? a = 5000, b =.07, = 8; = 5000(.07) 8 ; $6,899.66. The local animal rescue league is tring to reduce the number of stra dogs in the count. The estimate that there are currentl 00 stra dogs and that through their efforts the can place about 8% of the animals each month. How man stra dogs will remain in the count months after the animal control effort has started? a = 00, b = 0.9, = ; = 00(0.9) ; 7 stra dogs 3. A basket of groceries costs $96.50. Assuming an inflation rate of.8% per ear, how much will that same basket of groceries cost in 0 ears? a = 96.5, b =.08, = 0; = 96.5(.08) 0 ; $37.87
Eponential Growth and Deca While it is usuall fairl straightforward to determine a, the initial value, ou often need to read the problem carefull to make sure that ou are correctl identifing b and. This is especiall true when considering situations where the given growth rate is applied in intervals that are not the same as the given value of. You invest $000 in an account that pas 8% interest. If nothing changes, a = 000, b =.08, and = 3. The function is = 000 #.08 3. How much mone will ou have left after 3 ears? In this case, the interest is compounded annuall, meaning it is added to our account at the end of each ear. But what happens if the interest is compounded quarterl, meaning it is added to our account at the end of each quarter of a ear? There will be compounding periods in the 3-ear period. In each compounding period ou add the appropriate fraction of the total annual interest, in this case one-fourth of the interest. The new function is = 000 # a + 0.08, b given that a is the initial investment, b is the amount of growth for each compounding period, and is the number of compounding periods. Compounded annuall, the value of the investment will be $59.7 after 3 ears. Compounded quarterl, the value will be $68.. In Eercises and 5, identif a, b, and. Then write the eponential function that models the situation. Finall, use the function to answer the question.. You invest $000 in an investment that earns 6% interest, compounded quarterl. How much will the investment be worth after 5 ears? a = 000, b =.05, = 0; = 000(.05) 0 ; $693.7 5. You invest $3000 in an investment that earns 5% interest, compounded monthl. How much will the investment be worth after 8 ears? a = 3000, b =.0067, = 96; = 3000(.0067) 96 ; $7.89 6. The formula that financial managers and accountants use to determine the value of investments that are subject to compounding interest is A = P a + r nb nt where A is the final balance, P is the initial deposit, r is the annual interest rate, n is the number of times the interest is compounded per ear and t is the number of ears. Redo Eercises and 5 using this formula. Redo : A = P( + r n) nt = 000( + 0.06 ) # 5 = 000(.05) 0 = $693.7 Redo 5: A = P( + r n) nt = 3000( + 0.05 ) # 8 = 3000(.0067) 96 = $7.89
Solving Eponential Equations Eercises Solve each eponential equation.. 5 = 5 = 3. 5 = 5 = 3 3. 36 = 6 + 5 = 7. 3 = 6 = 5. Find the solution of the equation 8 = 3 - using both methods. a. Solve algebraicall. Eplain each step. 3 = 3 Rewrite 8 as a power with base. 3 = 3 Set the eponents equal to each other. = 3 = 3 Solve for. b. Solve graphicall. Justif our solution. 8 6 (, 8 ) 3 O The graphs intersect at the point ( 3, 8 ), so the solution is = 3.
Reteaching Geometric Sequences A geometric sequence is a sequence in which the ratio between consecutive terms is constant. This ratio is called the common ratio. Ever geometric sequence has a starting value and a common ratio. Is the following a geometric sequence?,,, 8, 6, c 8 6 3 3 3 3 The pattern is to multipl each term b to get the net term. Because there is a common ratio r =, the sequence is geometric. Determine whether the sequence is a geometric sequence. Eplain.. 5, 0, 0, 0, c. 3, 9, 7, 8, c es; the sequence has a es; the sequence has a common ratio of common ratio of 3 3.,, 9, 6, c., 0, 6,, c no; the sequence has no no; the sequence has a common ratio common difference 5. 5, 0, 5, 0, c 6. -8, 96, -9, 38, c no; the sequence has a common difference An geometric sequence can be defined b both an eplicit and a recursive definition. The recursive definition is useful for finding the net term in the sequence. a = a; a n = a n- # r In this formula, a represents the first term a n represents the nth term n represents the term number r represents the common ratio a n- represents the term immediatel before the nth term es; the sequence has a common ratio of
Geometric Sequences What is a recursive formula for the geometric sequence 3,, 8, 9, 768, c? 3 8 9 768 3 3 3 3 a = a; a n = a n- # r Use the recursive definition for a geometric sequence. a = 3; a n = a n- # Replace a with 3 and r with. Write the recursive formula for each geometric sequence. 7. 5, 5, 5, 65, c 8. -, 6, -8, 5, c a = 5; a n = a n ~5 a = ; a n = a n ~( 3) 9. 96, 7, 5, 0.5, c 0. 0, -0, 0, -0, c a a = 0; a n = a n ~( ) = 96; a n = a n ~ 3 An eplicit formula is more convenient when finding the nth term. What is an eplicit formula for the geometric sequence,,,, c? a n = a # r n- Use the eplicit definition for a geometric sequence. a n = - # (-) n- Replace a with - and r with -. Write the eplicit formula for each geometric sequence.. -3, 3, -3, 3, c., 0.5, 0.5, 0.5, c a n = 3 # ( ) n a n = # (0.5) n 3. 00, 00, 50, 5, c. 6,,, 8, c a n = 00 # (0.5) n a n = 6 # n
Reteaching Combining Functions Adding and Subtracting Functions You can add and subtract with functions just as ou do with numbers or epressions. Addition (f + g)() = f() + g() Subtraction (f g)() = f() g() Find the sum if f() = + 9 and g() = 5 + 6. What is (f g)(8)? Step Find (f - g)(). Step Evaluate (f - g)(8). (f - g)() = f() - g() (f - g)(8) = 7(8) + 3 = ( + 9) - (-5 + 6) = 59 = ( + 5) + (9-6) = 7 + 3 Find the sum or difference if f() = + 0 and g() = 8.. (f + g)(). (f + g)(9) 3. (f - g)(). (f - g)(9) (f + g)() = 3 + (f + g)(9) = 9 (f g)() = + 8 (f g)(9) = 9 A band s concert earnings are given b the equations below, where is the number of people in attendance. What function gives the band s total earnings? How much can the band epect to earn if 50 people attend the concert? Merchandise sales: M() =. Ticket sales: T() = 8 50 Step Find (M + T)(). Step Evaluate (M + T)(50). (M + T)() = M() + T() (M + T)(50) = 9.(50) - 50 = (.) + (8-50) = 60 = (. + 8) - 50 = 9. - 50 The band can epect to earn $60 if 50 people attend the concert. 5. The functions below give the number of car sales for two dealerships ears after opening. What is a function giving the difference in car sales between Dealership A and Dealership B? How man more cars is Dealership A epected to sell after 5 ears? Dealership A: A() = 35 + 650 Dealership B: B() = 0 + 5 (A B)() = 5 + 5; (A B)(5) = 300
Combining Functions Multipling and Dividing Functions You can multipl and divide with functions just as ou do with numbers or epressions. Multiplication (f # g)() = f() # f g() Division ag b() = f() and g() 0 g() Find the product if f() = 5 # 3 and g() = # 3. What is (f # g)()? Step Find (f # g)(). Step Evaluate (f # g)(). (f # g)() = f() # g() (f # g)() = 0 3 () = (5 3 ) # ( 3 ) = 0 # 3 = (5 # ) # (3 # 3 ) = 80 = 0 # 3 Find the product or quotient if f() = 3 + 9 and g() = 9. 6. (f # g)() 7. (f # g)() 8. a f g b() 9. a f gb() (f # g)() = 9 # 3 + 8 (f # g)() = 3 a f g b() = 3 9 + a f g b() = The number of orders a new restaurant epects to receive das after opening is O() = 0 + 5. Of those, D() = + 3 are epected to include dessert. Write a function P() giving the percentage of orders that include dessert. What percentage of orders is epected to include dessert after 0 das? Step Plan. Set up a function P() b multipling the ratio of dessert orders to food orders b 00. Then solve the equation for = 0 das. Step Find P(). P() = a D O b() # 00 = + 3 0 + 5 # 00 Step 3 Evaluate P(0). P(0) = (0) + 3 0(0) + 5 # 00 = 3 5 # 00 = 8. The restaurant can epect 8.% of orders to include dessert. 0. The population of an endangered animal in ears is modeled b f() = 0 # 0.8. The function g() =.7 represents the factor increase that a protection agenc uses to predict the change in population due to new protections. Write a function giving the population after ears. What is the predicted population after 3 ears? (f # g)() = 0 # 0.8 ; (f # g)(3)? 0
Reteaching Simplifing Radicals You can remove perfect-square factors from a radicand. What is the simplified form of 80n 5? In the radicand, factor the coefficient and the variable separatel into perfect square factors, and then simplif. Factor 80 and n 5 completel and then find paired factors. Solve 80 = 8 # 0 = # # # # 5 Factor 80 completel. = ( # )( # ) # 5 = ( # ) # 5 Find pairs of factors. 80 = # 5 = # 5 Use the rule ab = a # b. = # 5 = 5 The square root of a number squared is the number: a = a. n 5 = n # n # n # n # n Factor n 5 completel. = (n # n) # (n # n) # n = (n # n) # n Find pairs of factors. n 5 = (n # n) # n Separate the factors. = n # n = n n Remove the perfect square. 80n 5 = # n 5 # n = n 5n Combine our answers. Check 80n 5 n 5n Check our solution. 80n 5 n 5n 5n 5n 6n n Divide both sides b 5n. Simplif. n = n Solution: The simplified form of 80n 5 is n 5n. Eercises Simplif each radical epression.. 00n 3 0nn. 0b b 30 3. 66t 5 t 66t. 3 5. 55c 7 5c 3 c 6. 86t t86 7. 50g 3 5gg 8. 5h 6 3h 3 6 9. 35 35
Simplifing Radicals What is the simplified form of A 7t 8t? Begin b cancelling the common factors in the numerator and denominator. Simplif the numerator and denominator separatel when the denominator is a perfect square. Remember that the radical is not simplified if there is a radical in the denominator. Multipl to remove the radical from the denominator. Solve 7t 3 # A 8t = 5 3 3 # 3 # t # t # t 3 # # # t # t # t # t Factor the numerator and denominator completel. = 5 3 # 3 # 3 # t # t # t 3 # # # t # t # t # t Cancel the common factors. = (3 # 3) ( # )t = 3 t = 3 t = 3 (t) t (t) Find pairs of factors. These are the perfectsquare factors. Simplif the numerator and denominator separatel to remove the perfect-square factors. 3 = 3 and t = t Multipl the numerator and denominator b t to remove t from the denominator. = 3t t # = 3t = 3t t t t Remove the perfect-square factor from the denominator. 7t Solution: The simplified form of 3 3t A 8t is t. Eercises Simplif each radical epression. 0. 5 9 7 8. 8 9 A 00 3. 5a 5 A 9a 7 5 3a 3 0. 5 8s s 3. 0b 30b A b 3 5. 5 8 3 6t 6 7 s t 3 6. 50z 3 A 5zz 7. 56 t5 t t 8. 5 3t 8 t
Radical and Piecewise Functions +, for Ú - Graph the piecewise function f () = e - -, for 6 -. Step Make two tables. f() = + - - + = 0-3 - 3 + = - - + = - - + = 3 f() = - 5 -(-5) - = - 6 -(-6) - = - 7 -(-7) - = 3-8 -(-8) - = Step Plot the points on a graph. 6 O Eercises. Graph the function = - 3 8 + 5. What are the domain and range of the function? The -values -, 0,, ma help ou start a table. domain: all real numbers range: all real numbers O -, for Ú 3. Graph the piecewise function f () = e - +, for 6 3. 6 O 6 8