Exponential Functions Algebra III, Sec. 3.1 Objective Recognize, evaluate, and graph exponential functions.

Important Vocabulary  Algebraic functions – linear, polynomial, and rational functions  Transcendental functions – nonalgebraic functions (exponential & logarithmic functions)  Natural base e – …  Continuous compounding – the number of compoundings increase without bound

Exponential Functions The exponential function f with base a is denoted by ______________ where a > 0, a ≠ 1, and x is any real number.

Example (on your handout) Use a calculator to evaluate the expression

Example 1 Use a calculator to evaluate each function at the indicated value of x. a.) at x = π

Example 1 Use a calculator to evaluate each function at the indicated value of x. b.) at x = ½

Example 1 Use a calculator to evaluate each function at the indicated value of x. c.) at x =

Graphs of Exponential Fns For a > 1, the graph of is ___________________ over its domain. increasing decreasing

Graphs of Exponential Fns (cont.) For the graph of or, a > 1, the domain is ___________, the range is __________, and the y-intercept is __________. Also both graphs have _________ as a horizontal asymptote. (-∞, ∞) (0, ∞) (0, 1) y = 0

Example 2 In the same coordinate plane, sketch the graph of each function. a.) b.) Create a table. xf(x)g(x) Plot the points. Sketch the curve.

Example 3 In the same coordinate plane, sketch the graph of each function. a.) b.) Create a table. xf(x)g(x) Plot the points. Sketch the curve.

Example (on your handout) Sketch the graph of the function. Create a table. xf(x) Plot the points. Sketch the curve.

Exponents xx2x2 x3x3 x4x4 x5x

Example 4 Solve using the one-to-one property. a.) Write as common bases. Exponents must equal. Solve for x.

Example 4 Solve using the one-to-one property. b.) Write as common bases. Exponents must equal. Solve for x.

Example 5 Describe the graph as a transformation of the graph of a.) b.) c.) Horizontal shift: 3 units right Reflection over x-axis Reflection over y-axis, Vertical shift: 3 units up

The Natural Base e The natural exponential function is given by the function _______________. e = … is the constant x is the variable

Example 6 Use a calculator to evaluate the function given by at each individual value of x to four decimal places. a.) x = -0.4 b.) x = -7.1 c.) x =

Example (on your handout) Use a calculator to evaluate the expression

Example 7 Sketch the graph of the natural exponential function. Create a table. xf(x) Plot the points. Sketch the curve.

Applications After t years, the balance A in an account with principle P and annual interest rate r (in decimal form) is given by the formulas: For n compoundings per year: For continuous compounding:

Example (on your handout) Find the amount in an account after 10 years if $6000 is invested at an interest rate of 7%. a.) compounded monthly b.) compounded continuously

Example 8 On the day of a child’s birth, a deposit of $25,000 is made in a trust fund that pays 8.25% interest. Determine the balance in this account on the child’s 26 th birthday if the interest is compounded… a.) quarterly. b.) monthly. c.) continuously.