1. Geometric sequences can be modeled by exponential functions using the common ratio and the initial term.

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Brett Jackson
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1 1 Geometric sequences can be modeled by exponential functions using the common ratio and the initial term Exponential growth and exponential decay functions can be used to model situations where a quantity increases or decreases by the same rate in each time period The average rate of change for exponential functions either increases or decreases as the value of x increases Exponential functions can be used to predict the value at a particular point in time of the function 2 A population s growth can be modeled by an exponential growth function of the form where a > 0 and b > 0 If b > 1, then the value of the function increases as x increases 3 An exponential function repeatedly multiplies an initial amount by the same positive number, called the constant ratio The constant ratio is the factor that relates each value to the next 4 A geometric sequence is a number sequence formed by multiplying a term in the sequence by a fixed nonzero number, or a common ratio, to find the next term A sequence of numbers that is related by a common ratio is a geometric sequence 5 Compound interest is interest that is paid both on the principal and on the interest, that has already been paid Simple interest is paid only on the principal, and compound interest is paid on the principal and interest already earned 6 As x or y gets larger in absolute value, the graph of the exponential function gets closer to the line called an asymptote The graph of an exponential function approaches a straight line as x gets either very large or very small This line is the asymptote 7 8

9 10 11 The edge length of a cube with a volume of 64 is the cube root of 64 This can be expressed as,, or 4 12 Bicycle: 256 = 2 8 Luxury Vehicle: 65,536 = 2 16 =

2 The edge length of a cube with a volume of 64 is the cube root of 64 This can be expressed as,, or 4 12 Bicycle: 256 = 2 8 Luxury Vehicle: 65,536 = 2 16 = = The value of the luxury vehicle is 2 8 times the value of the bicycle 13 The graph passes through ( 2, 016), ( 1, 04), (0, 1), (1, 25), (2, 625)

3 14 The graph passes through ( 3, 0625), ( 2, 125), ( 1, 25), (0, 5), (1, 10), (2, 20) 15 Divide consecutive terms to find the constant ratio: 1 05 = 2, 2 1 = 2, 4 2 = 2; The constant ratio is 2 f(0) = 05, so the initial value is 05 The function is f(x) = 05(2) x

16 Divide consecutive terms to find the constant ratio: 7,500 2,500 = 3, 22,500 7,500 = 3, 67,500 22,500 = 3, 202,500 67,500 = 3; The constant ratio is 3 f(0) = 2,500, so the initial value is 2,500

4 16 Divide consecutive terms to find the constant ratio: 7,500 2,500 = 3, 22,500 7,500 = 3, 67,500 22,500 = 3, 202,500 67,500 = 3; The constant ratio is 3 f(0) = 2,500, so the initial value is 2,500 The function is f(x) = 2,500(3) x Evaluate the function for values of x to find the lowest value of x where f(x) > 45,000,000 f(8) = 16,402,500, f(9) = 49,207,500 The number of bacteria will exceed 45,000,000 in the 9 th month 17 To find the value of b, divide consecutive y values: = 6, 1, = 6; b = 6 18 Use the formula f(x) = ab x, where a is the initial value and b is the growth or decay factor f(x) = 50(115) x 19 Use the formula f(x) = ab x, where a is the initial value and b is the growth or decay factor f(x) = 200(085) x

5 20 Use the formula P = 12,000, r = 005, t = 10 When the interest is compounded twice per year, n = 2: When the interest is compounded annually, n = 1: 21 Use the formula P = 20,000, r = 0025, t = 15 When the interest is compounded four times per year, n = 4:, When the interest is compounded annually, n = 1: 22 Determine whether there is a common ratio between consecutive terms:,,, ; The common ratio is The initial value is 5 The explicit formula is The recursive formula is, 23 Determine whether there is a common ratio between consecutive terms: 5 2 = 25, 8 5 = 16; There is no common ratio, so the sequence is not a geometric sequence

6 24 Determine whether there is a common ratio between consecutive terms: 16 8 = 2, = 2, = 2, = 1; The common ratio is 2 The initial value is 8 The explicit formula is The recursive formula is, 25 Determine whether there is a common ratio between consecutive terms:,,, ; The common ratio is 2 The initial value is The explicit formula is The recursive formula is, 26 Identify the components of the geometric sequence from the explicit formula, r = 4 The recursive formula is, 27 Identify the components of the geometric sequence from the explicit formula, r = 35 The recursive formula is,

7 28 Find the common ratio by dividing consecutive terms: = 15, = 15, = 15; The common ratio is 15 The initial value is 40 The explicit formula is The recursive formula is, The end of the second week is day 14 The number of signatures will reach 7,000 by the end of the second week 29 Graphs of the form translate the graph of h units 0, and k units down when k < 0 So, the graph of is a translation of the graph of 5 units down 30 Graphs of the form translate the graph of h units 0, and k units down when k < 0 So, the graph of is a translation of the graph of 10 units up 31 Graphs of the form translate the graph of h units 0, and k units down when k < 0 So, the graph of is a translation of the graph of 2 units right 32 Graphs of the form translate the graph of h units 0, and k units down when k < 0 So, the graph of is a translation of the graph of 3 units left

g 2 units up from f 34 Graphs of the form translate the graph of h units 0, and k units down when k < 0 So, the graph of is a translation of

8 33 Graphs of the form translate the graph of h units 0, and k units down when k < 0 So, the graph of is a translation of the graph of 2 units up The graph of f passes through ( 1, 067), (0, 1), (1, 15), (2, 225), (3, 3375), (5, 76) (Note: some values are rounded) Graph f, then graph g 2 units up from f 34 Graphs of the form translate the graph of h units 0, and k units down when k < 0 So, the graph of is a translation of the graph of 05 unit down The graph of f passes through ( 2, 00625), ( 1, 025), (0, 1), (1, 4), (2, 16) Graph f, then graph g 05 units down from f

Unit 7 Exponential Functions. Name: Period:

Unit 7 Exponential Functions. Name: Period: Unit 7 Exponential Functions Name: Period: 1 AIM: YWBAT evaluate and graph exponential functions. Do Now: Your soccer team wants to practice a drill for a certain amount of time each day. Which plan will