Introduction An exponential function is a function in the form f(x) = a(b x ) + c, where a, b, and c are constants and b is greater than 0 but not equal to 1. An exponential expression is an expression that contains a base raised to a power/exponent. The expressions in exponential functions can often be rewritten using the properties of exponents in order to interpret information in real-world functions. For example, exponential functions can be used to model situations involving compound interest, or interest earned on both the initial amount and on previously earned interest. By rewriting these types of exponential functions, interest rates for different time periods can be determined : Rewriting Exponential Functions

Key Concepts In the exponential function form f(x) = a(b x ) + c, b x is an exponential expression; b represents the base, or the quantity that is being raised to a power in an exponential expression, and x represents the exponent or power, the quantity that shows the number of times the base is being multiplied by itself. If the base of an exponential function is greater than 1 (b > 1), the function models exponential growth, or an increase in an amount at a constant rate. Exponential growth can be represented by the formula f(t) = a(1 + r) t, where a is the initial value, (1 + r) is the growth factor, t is time, and f(t) is the final value : Rewriting Exponential Functions

Key Concepts, continued If the base of an exponential function is greater than 0 but less than 1 (0 < b < 1), the function models exponential decay, or a decrease in an amount at a constant rate. Exponential decay can be represented by the formula f(t) = a(1 – r) t, where a is the initial value, (1 – r) is the decay factor, t is time, and f(t) is the final value. An exponential function that has a base of e is called a natural exponential function. The value of e, an irrational number, is approximately : Rewriting Exponential Functions

Key Concepts, continued Natural exponential functions are commonly used in the fields of physics, chemistry, and business. Professionals in these fields often deal with situations involving continual growth or decay, such as analyzing the return on an investment or the progress of a chemical reaction. In banking, natural exponential functions are used to calculate compound interest that is being added to a balance at every instant. This is called continuously compounded interest. A single function has a change of rate that is dependent on the change that occurs between two points on the graph of that function. This is the same as the slope when the function is a line : Rewriting Exponential Functions

Key Concepts, continued The rate of change of a function is a ratio that describes how much one quantity changes with respect to the change in another quantity; mathematically, the rate of change can be represented by, in which f 1 (x) is the function value at the domain value x 1 and f 2 (x) is the function value at the domain value x 2. The uppercase Greek letter delta ( Δ ) is commonly used to represent the “change” in a value; for example, Δf(x) can be read as “change in f of x.” Therefore, Δf(x) and Δx are more concise ways of representing the numerator and denominator, respectively, in the rate of change formula : Rewriting Exponential Functions

Key Concepts, continued Note that the order in which the function values are compared must be the same as the order in which the domain values are compared. Mixing up the order of the values in the numerator with the order of the values in the denominator will result in an incorrect rate calculation. The percent rate of change of a function is the percentage by which the function increases or decreases within a certain interval. If this percent is positive, then the function shows exponential growth; if the percent is negative, then the function shows exponential decay. The following table of properties can be helpful when rewriting exponential functions : Rewriting Exponential Functions

Key Concepts, continued : Rewriting Exponential Functions Properties of Exponents PropertyGeneral Rule Zero Exponent Propertya 0 = 1 Negative Exponent Property Product of Powers Property Quotient of Powers Property (continued)

Key Concepts, continued : Rewriting Exponential Functions Properties of Exponents PropertyGeneral Rule Power of a Power Property Power of a Product Property Product of a Quotient Property Rational Exponents Property

Common Errors/Misconceptions adding exponents instead of multiplying them when raising a power to a power forgetting to multiply by 100 when converting a decimal to a percent not realizing that a rate written as a decimal is added to 1 in a function that models exponential growth not realizing that a rate written as a decimal is subtracted from 1 in a function that models exponential decay : Rewriting Exponential Functions

Guided Practice Example 3 Meg has a $5,000 student loan. For the next x years she is paying only the interest, which accumulates at a rate of 8% per year after monthly compounding. The function can be used to find the approximate monthly interest rate of Meg’s loan, where n is the number of payments made each year (12). What is the approximate monthly interest rate? : Rewriting Exponential Functions

Guided Practice: Example 3, continued 1.Evaluate the function for n = 12. This will yield a function that accounts for monthly interest. Original function Substitute 12 for n. f(x) ≈ 5000(1.0064) 12x Use a calculator to simplify : Rewriting Exponential Functions

Guided Practice: Example 3, continued 2.Subtract 1 from the number inside the parentheses and convert the difference to a percent. The number in parentheses is the growth factor of the monthly interest on the original loan amount. To determine the growth rate, subtract 1 from this amount – 1 = : Rewriting Exponential Functions

Guided Practice: Example 3, continued Multiply the result by 100 to convert the interest rate to a percentage = 0.64% The approximate monthly interest rate of the loan is 0.64% : Rewriting Exponential Functions ✔

Guided Practice: Example 3, continued : Rewriting Exponential Functions

Guided Practice Example 4 Eton originally purchased his car for $16,000, but since then the car has been depreciating, or losing value, at a rate of 12% per year. The exponential function can be used to determine the value of Eton’s car at the end of each month, where n is the number of times per year that the car’s value is being determined. In this case, n is 12. What is the car’s approximate monthly depreciation rate? : Rewriting Exponential Functions

Guided Practice: Example 4, continued 1.Evaluate the function for n = 12. This will yield a function that describes the car’s monthly depreciation. Original function Substitute 12 for n. v(x) ≈ 16,000(0.9894) 12 Use a calculator to simplify : Rewriting Exponential Functions

Guided Practice: Example 4, continued 2.Subtract the number inside the parentheses from 1 and convert the differences to a percent. The number in parentheses is the decay factor of the car’s value each month. To determine the decay rate, subtract this amount from 1. Notice that we are subtracting from 1 here since this function represents decay. 1 – = : Rewriting Exponential Functions

Guided Practice: Example 4, continued Multiply the result by 100 to convert the interest rate to a percentage = 1.06% The car’s approximate monthly depreciation rate is 1.06% : Rewriting Exponential Functions ✔

Guided Practice: Example 4, continued : Rewriting Exponential Functions