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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. x 1 x Graph y1 = 2 and y2 = 2 When b > 1, the function f(x) = bx represents exponential growth. When 0 < b < 1, the function f(x) = bx represents exponential decay. Graph f(x) = 2x along with each function below. Tell whether each function represents exponential growth or exponential decay. Then give the y-intercept. a) 4 f(x) 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 x 1 x b) 6 f( x) y 6 2 6 2 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 The total amount of an investment, A, earning compound interest is nt r A(t) P 1 , n 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. Find the final amount of a $500 investment after 8 years at 7% interest compounded annually, quarterly, and monthly. nt r A(t) P 1 n 18 0.07 A(t) 500 1 compounded annually: = $859.09 1 4 8 0.07 compounded quarterly: A(t) 500 1 = $871.11 4 128 0.07 compounded monthly: A(t) 500 1 12 = $873.91 Find the final amount of a $2200 investment at 9% interest compounded monthly for 3 years. 1. The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers. 2. The graphs of all exponential functions pass through the point (0, 1) because f (0) = b0 = 1. 3. If b > 1, f (x) = bx has a graph that goes up to the right and is an increasing function. 4. If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a decreasing function. 5. f (x) = bx is a one-to-one function and has an inverse that is a function. 6. The graph of f (x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote. f (x) = bx f (x) = bx 0<b<1 b>1 Transformation Equation Description Horizontal translation g(x) = bx+c • Shifts the graph of f (x) = bx to the left c units if c > 0. Vertical stretching or shrinking g(x) = c bx 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. Reflecting g(x) = -bx g(x) = b-x • Reflects the graph of f (x) = bx about the x-axis. g(x) = -bx + c • Shifts the graph of f (x) = bx upward c units if c > 0. Vertical translation • Shifts the graph of f (x) = bx to the right c units if c < 0. • Reflects the graph of f (x) = bx about the y-axis. • Shifts the graph of f (x) = bx downward c units if c < 0. 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 (-1, 1) -5 -4 -3 -2 -1 (0, 1) 1 2 3 4 5 6 Sketch a graph using transformation of the following: 1. f ( x) 2 3 2. f ( x) 2 x 1 3. x f ( x) 4 x 1 1 Recall the order of shifting: horizontal, reflection (horz., vert.), vertical. 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) = 3x f (x) = ex e 2.71828... 4 f (x) = 2x (1, 3) 3 (1, e) 2 (1, 2) (0, 1) 1 -1 For continuous compounding: A= rt Pe Must pass the horizontal line test. Is this function one to one? Yes Does it have an inverse? Yes 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. log381 = 4 34 = 81 log168 = 3/4 163/4 = 8 Write the exponential equation in logarithmic form 82 = 64 log 8 64 = 2 4-3 = 1/64 log4 (1/64) = -3 f(x) = log232 f(x) = log42 4y = 2 22y = 21 y = 1/2 f(x) = log31 3y = 1 y=0 Think: y = log232 Step 1- rewrite it as an exponential equation. y = 132 2 f(x) = log10( /100) Step 2- make the bases the 10ysame. = 1/100 10y = 2 10y-2= 25 y = -2 Therefore, y=5 You can only use a calculator when the base is 10 Find the log key on your calculator. Evaluate the following using that log key. log 10 = 1 log 1/3 = -.4771 log 2.5 = .3979 log -2 = ERROR!!! Why? loga1 0 a = 0 because = 1 logaa = 1 because a1 = a logaax = x and alogax = x If logax = logay, then x = y Rewrite as an exponent 4y = 1 Therefore, y = 0 log41= 0 log77 = 1 log620 6 Rewrite as an exponent y 7 = 7 Therefore, y = 1 = 20 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 Rewrite the x = 3y exponential as a y = log3x logarithm… f-1(x) = log3x Find the inverse of the following exponential functions… f(x) = 2x f(x) = 2x+1 f(x) = 3x- 1 f-1(x) = log2x f-1(x) = log2x - 1 f-1(x) = log3(x + 1) Find the inverse of the following logarithmic functions… f(x) = log4x f(x) = log2(x - 3) f(x) = log3x – 6 f-1(x) = 4x f-1(x) = 2x + 3 f-1(x) = 3x+6 Graphs of Logarithmic Functions It is the inverse of y = 3x x y = 3 y= log3x Therefore, the Domain? table of values for (0,) x y x y g(x) will-1be the 1/ 1/ -1 3 Range?3 (-,) reverse of the 0 1 1 0 table of values for Asymptotes? x=0 3 1 y = 31x. 3 2 9 9 2 Graphs of Logarithmic Functions g(x) = log4(x – 3) What is the inverse exponential function? y= 4x + 3 Show your tables of values. y= log4(x – 3) y= 4x + 3 Domain? (3,) x y x y 3.25 -1 -1Range? 3.25(-,) 4 0 0 4 Asymptotes? x = 3 7 1 1 7 19 2 2 19 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 y= log5(x – 1) + 4 x xDomain? y (1,) 1.2 3Range? 1.2 (-,) 2 4 2 Asymptotes? x = 1 6 5 6 26 6 26 y 3 4 5 6 The function defined by f(x) = logex = ln x, x > 0 is called the natural logarithmic function. Find the ln key on your calculator. Evaluate the following using that ln key. ln 2 = .6931 ln 7/8 = -.1335 ln 10.3 = ln -1 2.3321 = ERROR!!! Why? 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 ln 1/e= -1 2 ln e = 2 5 ln e = 5 Rewrite as an exponent ey = 1/e ey = e-1 Therefore, y = -1 Rewrite as an exponent ln e = y/2 e y/2 = e1 Therefore, y/2 = 1 and y = 2. Graphs of Natural Log Functions g(x) = ln(x + 2) Show your table of values. y= ln(x + 2) x y -2 error Domain? (-2,) -1 0 0 (-,) .693 Range? 1 1.099 Asymptotes? x = -2 2 1.386 Graphs of Natural Log Functions g(x) = ln(2 - x) Show your table of values. y= ln(2 - x) x y 2 error Domain? (-2,) 1 0 0 (-,) .693 Range? -1 1.099 Asymptotes? x = -2 -2 1.386