Exponential Growth and Decay

Identifying Exponential Growth vs Decay Exponential Equation: f(x) = Cax C: COEFFICIENT a: BASE X: EXPONENT Exponential Growth When the base is greater than 1, a > 1 f(x) is INCREASING Exponential Decay When the base is between zero and one, 0 < a < 1 f(x) is DECREASING 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 1 Determine whether represents exponential growth or decay. 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 2 Determine whether represents exponential growth or decay. 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 3 Determine whether represents exponential growth or decay. 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

Graphing Exponentials Make a table of values, usually from [–3, 3] Plot the points from the table Connect the dots and draw from left to right, a smooth curve and label any asymptotes 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 4 Graph f(x) = 2x x f(x) –3 (2)–3 1/8 –2 (2)–2 1/4 –1 (2)–1 1/2 (2)0 1 (2)1 2 (2)2 4 3 (2)3 8 x f(x) –3 (2)–3 –2 (2)–2 –1 (2)–1 (2)0 1 (2)1 2 (2)2 3 (2)3 x f(x) –3 –2 –1 1 2 3 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 4 Graph f(x) = 2x x f(x) –3 (2)–3 1/8 –2 (2)–2 1/4 –1 (2)–1 1/2 (2)0 1 (2)1 2 (2)2 4 3 (2)3 8 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 4 Graph f(x) = 2x (Calculator check) 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 5 Graph x f(x) –3 –2 –1 1 2 3 –3 (1/3)(3)–3 1/81 –2 (1/3)(3)–2 1/27 –1 (1/3)(3)–1 1/9 (1/3)(3) 0 1/3 1 (1/3)(3) 1 2 (1/3)(3) 2 3 (1/3)(3) 3 9 –3 (1/3)(3)–2 –2 –1 (1/3)(3)–1 (1/3)(3) 0 1 (1/3)(3) 1 2 (1/3)(3) 2 3 (1/3)(3) 3 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 6 Graph x f(x) –3 –2 –1 1 2 3 –3 (1/2)(4)–3 1/128 –2 (1/2)(4)–2 1/32 –1 (1/2)(4)–1 1/8 (1/2)(4) 0 ½ 1 (1/2)(4) 1 2 (1/2)(4) 2 8 3 (1/2)(4) 3 32 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 6 Graph x f(x) –3 (1/2)–3 8 –2 (1/2)–2 4 –1 (1/2)–1 2 (1/2)0 1 (1/2)1 1/2 (1/2)2 1/4 3 (1/2)3 1/8 x f(x) –3 (1/2)–3 –2 (1/2)–2 –1 (1/2)–1 (1/2)0 1 (1/2)1 2 (1/2)2 3 (1/2)3 x f(x) –3 –2 –1 1 2 3 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 6 Graph x f(x) –3 (1/2)–3 8 –2 (1/2)–2 4 –1 (1/2)–1 2 (1/2)0 1 (1/2)1 1/2 (1/2)2 1/4 3 (1/2)3 1/8 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

Growth/Decay Factor Equation Equation: A = P (1 + r)t P: Initial Principle +: Growth Factor -: Decay Factor r: Interest Rate t: Time it takes to accrue amount 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 7 You invest $5,000 in an account that pays 10% interest per year. How much money will your investment be in 3 years? A = ? Do you know much you are going to make? P = $5,000 $5,000 is deposited r = 0.10 Interest Rate – remember it needs to be in decimal form t = 3 Time it takes to accrue amount + Growth 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 7 You invest $5,000 in an account that pays 10% interest per year. How much money will your investment be in 3 years? 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 8 You buy a car that cost $5,000 and depreciates 10% per year. What is the value of the car after 3 years? A = ? Do you know much you are going to make? P = $5,000 $5,000 is spent r = 0.10 Interest Rate – remember it needs to be in decimal form t = 3 Time it takes to accrue amount Decay Decreasing Value 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 8 You buy a car that cost $5,000 and depreciates 10% per year. What is the value of the car after 3 years? 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 9 You buy a car that cost $10,000 and depreciates 10% per year. How much money have you lost, compared to the original amount, in 5 years? 10/28/2019 1:24 AM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 - Exponential Functions Example 10 You invest $5000 in an account that pays 6.25% interest per year. How much money will your investment be in 5 years? 10/28/2019 1:24 AM 7.1 - Exponential Functions

7.1 - Exponential Functions Example 10 You invest $5000 in an account that pays 6.25% interest per year. How much money will your investment be in 5 years? A = ? Do you know much you are going to make? P = $5,000 $5,000 is deposited r = 6.25% Interest Rate – remember it needs to be in decimal form t = 5 Time it takes to accrue amount 10/28/2019 1:24 AM 7.1 - Exponential Functions

7.1 - Exponential Functions Example 10 You invest $5000 in an account that pays 6.25% interest per year. How much money will your investment be in 5 years? Plug into the equation 10/28/2019 1:24 AM 7.1 - Exponential Functions

7.1 - Exponential Functions Example 11 You buy a car that cost $5,000 and depreciates 6.25% per year. How much money will you end up paying in 5 years? 10/28/2019 1:24 AM 7.1 - Exponential Functions

7.1 - Exponential Functions Example 11 You buy a car that cost $5,000 and depreciates 6.25% per year. How much money will you end up paying in 5 years? A = ? Do you know much you are going to pay? P = $5,000 $5,000 is borrowed r = 6.25% Interest Rate – remember it needs to be in decimal form t = 5 Time it takes to accrue amount 10/28/2019 1:24 AM 7.1 - Exponential Functions

7.1 - Exponential Functions Example 11 You buy a car that cost $5,000 and depreciates 6.25% per year. How much money will you end up paying in 5 years? Plug into the equation 7.1 - Exponential Functions 10/28/2019 1:24 AM

7.1 - Exponential Functions Example 12 The value of a $3000 computer decreases about 30% each year. Write a function for the computer’s value in 4 years. Does the function represent growth or decay? 10/28/2019 1:24 AM 7.1 - Exponential Functions

7.1 - Exponential Functions Example 12 The value of a $3000 computer decreases about 30% each year. Write a function for the computer’s value in 4 years. Does the function represent growth or decay? 10/28/2019 1:24 AM 7.1 - Exponential Functions

7.1 - Exponential Functions Time Values Annually is a one-time payment per year Semiannually/Biannually is a payment every six months (2 times a year) Quarterly is a payment every three months (4 times a year) Monthly is a payment every month (12 times a year) Daily is a payment every day (365 times a year) 10/28/2019 1:24 AM 7.1 - Exponential Functions

Compound Interest Equation A = Total Amount Earned P = Principal r = Interest Rate n = Compounded Amount t = Time 10/28/2019 1:24 AM 7.1 - Exponential Functions

7.1 - Exponential Functions Example 13 $5,000 is deposited in an account that pays 6% annual interest compounded quarterly. Find the balance after 25 years if the interest is compounded quarterly. A = ? Do we know how much it is when the balance after 25 years? P = $5,000 $5,000 is deposited r = 6% Interest Rate – remember it needs to be in decimal form n = 4 Compounded quarterly t = 25 Time it takes to accrue amount 10/28/2019 1:24 AM 7.1 - Exponential Functions

7.1 - Exponential Functions Example 13 $5,000 is deposited in an account that pays 6% annual interest compounded quarterly. Find the balance after 25 years if the interest is compounded quarterly. 10/28/2019 1:24 AM 7.1 - Exponential Functions

7.1 - Exponential Functions Example 13 Assume $5,000 is deposited in an account that pays 6% annual interest compounded quarterly. Find the balance after 25 years if the interest is compounded quarterly. Plug into the equation 10/28/2019 1:24 AM 7.1 - Exponential Functions

7.1 - Exponential Functions Example 14 How much must you deposit in an account that pays 6.5% interest, compounded quarterly, to have a balance of $5,000 in 15 years? 10/28/2019 1:24 AM 7.1 - Exponential Functions

7.1 - Exponential Functions Example 14 How much must you deposit in an account that pays 6.5% interest, compounded quarterly, to have a balance of $5,000 in 15 years? 10/28/2019 1:24 AM 7.1 - Exponential Functions