E-5. 3: General Form of the Exponential Function E-5 E-5.3: General Form of the Exponential Function E-5.4: Building functions in the form 𝑓 𝑥 = 𝑎 𝑥

Answers to page: 131 1.a. P(0) = 1000 b. P(1) = 2000 P(2)=4000 P(3)=8000 c. Constant ratio = 2. It is an exponential function because you multiply to get to the next output d. Growth. The constant ratio is greater than one. Also, we can see that the values are getting larger

Answers to page: 132 2.a. see graph b. W(4)=16 W(5) = 8 W(6)=4 c. Constant ratio = 0.5. This is an exponential function because you are multiplying by 0.5 to get to the next output d. It is decay. The constant ratio is 0<a<1 and we can see that the values are getting smaller

Answers to page: 133 3. a. At month zero (or at the beginning) you have $100 b. At 12 months you have $313.84 c. See table d. The constant ratio is 1.1. It is an exponential function because you multiply to get to the next output

Answers to page: 134 4. a. When you bought the car it is worth $20,000 b. At 4 years (or 4 years after you bought the car) the car is worth $10,440 c. 100 – 15% = 85%. If it loses 15% then it must retain 85% d. See table e. Exponential decay. The constant ratio is 0.85 which is 0<a<1 and we can see that the outputs are getting smaller

Answers to page: 137

Answers to page: 137

Answers to page: 138

Answers to page: 138

Answers to page: 143-144 Constant ratio = 2 2. a. 1 represents the initial value b. The number of cuts tells you how many factors of 2 that need to be multiplied by to determine the number of pieces you have 3. 1∗ 2 𝑛 4. P(10)= 1∗ 2 10 =1024

Answers to page: 145 5. 6. 𝐴 𝑛 =1∗ 1 2 𝑛 A(9)=1∗ 1 2 9 = 1 512

Essential Question: How do I create an exponential function? I CAN: Find the constant ratio given two points in an exponential function Find the initial value of an exponential function when given at least one point and the constant ratio Substitute the constant ratio and initial value to build the symbolic representation of an exponential function

Reminder: How do I find constant ratio? Constant ratio = 𝑜𝑢𝑡𝑝𝑢𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑜𝑢𝑡𝑝𝑢𝑡 Example: (2, 12) and (3, 36) Example: X F(x) 4 5 2

Reminder: How do I find constant ratio? Constant ratio = 𝑜𝑢𝑡𝑝𝑢𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑜𝑢𝑡𝑝𝑢𝑡 Example: (2, 12) and (3, 36) Constant Ratio = 36 12 =3 Example: Constant Ratio = 2 4 =0.5 X F(x) 4 5 2

How do I find the initial value? Multiply by the constant ratio to get future outputs, divide by the constant ratio to get previous outputs Example: (2, 12) and (3, 36) Example: X F(x) 1 2 3 4 5

How do I find the initial value? Multiply by the constant ratio to get future outputs, divide by the constant ratio to get previous outputs Example: (2, 12) and (3, 36) Constant Ratio = 36 12 =3 (1, 4) and (0, 4 3 ) Example: Constant Ratio = 2 4 =0.5 X F(x) 64 1 32 2 16 3 8 4 5

How do I build the symbolic representation? Substitute the constant ratio for “a” and the initial value for “c” into the equation: 𝑓 𝑥 =𝑐∗ 𝑎 𝑥 Example: (2, 12) and (3, 36) Example: X F(x) 1 2 3 4 5

How do I build the symbolic representation? Substitute the constant ratio for “a” and the initial value for “c” into the equation: 𝑓 𝑥 =𝑐∗ 𝑎 𝑥 Example: (2, 12) and (3, 36) 𝑓 𝑥 = 4 3 ∗ 3 𝑥 Example: 𝑓 𝑥 =64∗ 0.5 𝑥 X F(x) 1 2 3 4 5