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Exponentials Initial value Growth or decay factor Re-call exponential form is 𝑦=𝑎∗ 𝑏 𝑥 , a≠0, and b>0 𝑎 represents: Initial value 𝑏 represent the Growth or decay factor

Is it growth or decay? For each I.D. growth or decay, initial value and growth/decay factor: 1) 𝑦=8 3 𝑥 2) 𝑦=2 1 2 𝑥 3) 𝑦= 1 3 2 𝑥 4) 𝑦=0.16 0.25 𝑥

Answers Growth, a= 8 and factor is b= 3 Decay, a = 2 and factor is b= ½ Growth a= 1/3 and factor is b=2 Decay a=0.16 and factor is b=0.25

Good Morning Exponentials

Sketching a graph Make a table of values: 𝑦= 3 𝑥 x y -1 1 2 3 1/3 1 3 1 2 3 1/3 1 3 9 27

Plot values to get the graph

Finding equations Given points: 0,1 𝑎𝑛𝑑 2, 36 Find the equations by substituting points in and solving for “a” and “b”

By Hand Use equation 𝑦=𝑎∗ 𝑏 𝑥 to solve for a. when x=0 then y=1. So 1=𝑎∗ 𝑏 0 well 𝑏 0 =1 so 𝑎=1 Now use second point and solve for b. 36=1∗ 𝑏 2 So 𝑏=±6 but we know 𝑏>0 so B=6

SO our equation is 𝑦=1∗ 6 𝑥 You try: write the exponential from the point given 0,1 𝑎𝑛𝑑 2, 49 0, 3 𝑎𝑛𝑑 2, 12 1, 9 (4, 72)

Solutions 𝑦= 5 𝑥 𝑦=3 ∗ 2 𝑥 Hmmm how do we do that one by hand? We get stuck with 9=𝑎∗ 𝑏 1 𝑎𝑛𝑑 72=𝑎∗ 𝑏 4

Steps to do it in the calculator given points/ data Stat → Edit → 𝑒𝑛𝑡𝑒𝑟 𝑑𝑎𝑡𝑎 𝑖𝑛 𝐿 1 𝑎𝑛𝑑 𝐿 2 𝑆𝑡𝑎𝑡 →𝐶𝑎𝑙𝑐→𝐸𝑥𝑝𝑅𝑒𝑔 𝑦=𝑎∗ 𝑏 𝑥 𝑎=𝑠𝑜𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑏=𝑠𝑜𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟

Recall parent function Transformations If we had the function 𝑦= 𝑥 2 How did we describe the transformation(s): 𝑦= 𝑥−3 2 𝑦= 𝑥 2 +4 𝑦=− 𝑥 2 𝑦= −𝑥 2

The same translations apply to exponential functions We can use the characteristic point we know for exponentials to help use I.D. graphs The point is 0,1

Examples 𝑦= 2 𝑥 𝑝𝑜𝑖𝑛𝑡 𝑜𝑛 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑖𝑠 0,1 𝑎𝑛𝑑 2,4 If we transform the graph: 𝑦= 2 𝑥+1 the graph has been shifted one unit to the left so instead of 0,1 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑖𝑠 𝑛𝑜𝑤 −1, 1

Try it 𝑦= 2 𝑥 Describe the translation(s) use the point 0,1 as a reference. 𝑦= 2 𝑥 +3 𝑦=− 2 𝑥 𝑦= 2 −𝑥 𝑦= 2 𝑥−3 +4