Exponential Functions

Exponential Functions y = x2 y = 2x The function f(x) = bx is an exponential function with base b, where b is a positive real number other than 1 and x is any real number. An asymptote is a line that a graph approaches (but does not reach) as its x- or y-values become very large or very small.

Exponential Functions Graph y1 = 2x and y2 = When b > 1, the function f(x) = bx represents exponential growth. When 0 < b < 1, the function f(x) = bx represents exponential decay.

Example 1 Graph f(x) = 2x along with each function below. Tell whether each function represents exponential growth or exponential decay. Then give the y-intercept. y = 4(2x) exponential growth, since the base, 2, is > 1 y-intercept is 4 because the graph of f(x) = 2x, which has a y-intercept of 1, is stretched by a factor of 4 exponential decay, since the base, ½, is < 1 y-intercept is 6 because the graph of f(x) = 2x, which has a y-intercept of 1, is stretched by a factor of 6

Compound Interest The total amount of an investment, A, earning compound interest is where P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years.

Example 1 Find the final amount of a $500 investment after 8 years at 7% interest compounded annually, quarterly, and monthly. compounded annually: = $859.09 compounded quarterly: = $871.11 compounded monthly: = $873.91

Practice Find the final amount of a $2200 investment at 9% interest compounded monthly for 3 years.

Characteristics of Exponential Functions The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers. The graphs of all exponential functions pass through the point (0, 1) because f (0) = b0 = 1. If b > 1, f (x) = bx has a graph that goes up to the right and is an increasing function. If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a decreasing function. f (x) = bx is a one-to-one function and has an inverse that is a function. The graph of f (x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote. f (x) = bx b > 1 f (x) = bx 0 < b < 1

Transformations Involving Exponential Functions Shifts the graph of f (x) = bx upward c units if c > 0. Shifts the graph of f (x) = bx downward c units if c < 0. g(x) = -bx + c Vertical translation Reflects the graph of f (x) = bx about the x-axis. Reflects the graph of f (x) = bx about the y-axis. g(x) = -bx g(x) = b-x Reflecting Multiplying y-coordintates of f (x) = bx by c, Stretches the graph of f (x) = bx if c > 1. Shrinks the graph of f (x) = bx if 0 < c < 1. g(x) = c bx Vertical stretching or shrinking Shifts the graph of f (x) = bx to the left c units if c > 0. Shifts the graph of f (x) = bx to the right c units if c < 0. g(x) = bx+c Horizontal translation Description Equation Transformation

Example Use the graph of f (x) = 3x to obtain the graph of g(x) = 3 x+1. Solution Examine the table below. Note that the function g(x) = 3x+1 has the general form g(x) = bx+c, where c = 1. Because c > 0, we graph g(x) = 3 x+1 by shifting the graph of f (x) = 3x one unit to the left. We construct a table showing some of the coordinates for f and g to build their graphs. f (x) = 3x g(x) = 3x+1 (0, 1) (-1, 1) 1 2 3 4 5 6 -5 -4 -3 -2 -1

Problems Sketch a graph using transformation of the following: 1. 2. 3. Recall the order of shifting: horizontal, reflection (horz., vert.), vertical.

The Natural Base e An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. More accurately, The number e is called the natural base. The function f (x) = ex is called the natural exponential function. f (x) = ex f (x) = 2x f (x) = 3x (0, 1) (1, 2) 1 2 3 4 (1, e) (1, 3) -1

Formulas for Compound Interest For continuous compounding: A = Pert

Logarithmic Functions & Their Graphs

Is this function one to one? Must pass the horizontal line test. f(x) = 3x Is this function one to one? Yes Does it have an inverse? Yes

Logarithmic Function of base “a” Definition: Logarithmic function of base “a” - For x > 0, a > 0, and a  1, y = logax if and only if x = ay Read as “log base a of x” f(x) = logax is called the logarithmic function of base a.

The most important thing to remember about logarithms is…

a logarithm is an exponent.

Write the logarithmic equation in exponential form 34 = 81 163/4 = 8 Write the exponential equation in logarithmic form 82 = 64 4-3 = 1/64 log 8 64 = 2 log4 (1/64) = -3

Evaluating Logs 2y = 32 2y = 25 y = 5 Think: y = log232 f(x) = log232 Step 1- rewrite it as an exponential equation. f(x) = log42 4y = 2 22y = 21 y = 1/2 2y = 32 f(x) = log10(1/100) Step 2- make the bases the same. 10y = 1/100 10y = 10-2 y = -2 2y = 25 f(x) = log31 Therefore, y = 5 3y = 1 y = 0

Evaluating Logs on a Calculator You can only use a calculator when the base is 10 Find the log key on your calculator.

Why? log 10 = 1 log 1/3 = -.4771 log 2.5 = .3979 log -2 = ERROR!!! Evaluate the following using that log key. log 10 = 1 log 1/3 = -.4771 log 2.5 = .3979 log -2 = ERROR!!! Why?

Properties of Logarithms loga1 = 0 because a0 = 1 logaa = 1 because a1 = a logaax = x and alogax = x If logax = logay, then x = y

Simplify using the properties of logs Rewrite as an exponent 4y = 1 Therefore, y = 0 log41= log77 = 1 Rewrite as an exponent 7y = 7 Therefore, y = 1 6log620 = 20

Use the properties of logs to solve these equations. log3x = log312 x = 12 log3(2x + 1) = log3x 2x + 1 = x x = -1 log4(x2 - 6) = log4 10 x2 - 6 = 10 x2 = 16 x = 4

Review: How do you find the inverse of a function? Application of what you know… What is the inverse of f(x) = 3x? y = 3x x = 3y y = log3x f-1(x) = log3x Rewrite the exponential as a logarithm…

Find the inverse of the following exponential functions… f(x) = 2x f-1(x) = log2x f(x) = 2x+1 f-1(x) = log2x - 1 f(x) = 3x- 1 f-1(x) = log3(x + 1)

Find the inverse of the following logarithmic functions… f(x) = log4x f-1(x) = 4x f(x) = log2(x - 3) f-1(x) = 2x + 3 f(x) = log3x – 6 f-1(x) = 3x+6

Graphs of Logarithmic Functions Graph g(x) = log3x It is the inverse of y = 3x Therefore, the table of values for g(x) will be the reverse of the table of values for y = 3x. y = 3x x y -1 1/3 1 3 2 9 y= log3x x y 1/3 -1 1 3 9 2  Domain? (0,)   Range? (-,)  Asymptotes? x = 0

Graphs of Logarithmic Functions g(x) = log4(x – 3) What is the inverse exponential function? y= 4x + 3 Show your tables of values.    y= 4x + 3 x y -1 3.25 4 1 7 2 19 y= log4(x – 3) x y 3.25 -1 4 7 1 19 2 Domain? (3,) Range? (-,) Asymptotes? x = 3

Graphs of Logarithmic Functions g(x) = log5(x – 1) + 4 What is the inverse exponential function? y= 5x-4 + 1  Show your tables of values.   y= 5x-4 + 1 x y 3 1.2 4 2 5 6 26 y= log5(x – 1) + 4 x y 1.2 3 2 4 6 5 26 Domain? (1,) Range? (-,) Asymptotes? x = 1

Natural Logarithmic Functions The function defined by f(x) = logex = ln x, x > 0 is called the natural logarithmic function.

Evaluating Natural Logs on a Calculator Find the ln key on your calculator.

Why? ln 2 = .6931 ln 7/8 = -.1335 ln 10.3 = 2.3321 ln -1 = ERROR!!! Evaluate the following using that ln key. ln 2 = .6931 ln 7/8 = -.1335 ln 10.3 = 2.3321 ln -1 = ERROR!!! Why?

Properties of Natural Logarithms ln1 = 0 because e0 = 1 Ln e = 1 because e1 = e ln ex = x and eln x = x If ln x = ln y, then x = y

Use properties of Natural Logs to simplify each expression Rewrite as an exponent ey = 1/e ey = e-1 Therefore, y = -1 ln 1/e= -1 2 ln e = 2 Rewrite as an exponent ln e = y/2 e y/2 = e1 Therefore, y/2 = 1 and y = 2. 5 eln 5=

Graphs of Natural Log Functions g(x) = ln(x + 2) Show your table of values. y= ln(x + 2) x y -2 error -1 .693 1 1.099 2 1.386   Domain? (-2,)   Range? (-,) Asymptotes? x = -2

Graphs of Natural Log Functions g(x) = ln(2 - x) Show your table of values. y= ln(2 - x) x y 2 error 1 .693 -1 1.099 -2 1.386   Domain? (-2,)   Range? (-,) Asymptotes? x = -2