(a) (b) (c) (d) Warm Up: Show YOUR work!

Warm Up

Warn Up

Section 5.2/5.4 Exponential and Logarithmic Functions p. 339: (even), 37 p. 358: (even), 31-38

Definition of the Exponential Function  The exponential function f with base b is defined by f(x) =b x or y = b x  Where “b” is a positive constant other than 1 and greater than 0. YESf(x) = 2 x f(x) = 10 x f(x) = 3 x+1 f(x) = (1/2) x-1 NOf(x) = x 2 f(x) = 1 x f(x) = (-1) x f(x) = x x

Exponential Growth vs. Decay: y = b x Growth  b>1  Domain: (-∞,∞)  Range: (0,∞)  y-intercept is (0,1)  As x increases, for b > 1, f(x) also increases without bound  The x-axis (y = 0) is the asymptote Decay  0<b<1  Domain: (-∞,∞)  Range: (0,∞)  y-intercept is (0,1)  As x increases, for 0<b<1, f(x) decreases, approaching zero  The x-axis (y = 0) is the asymptote

Example

Consider the function y = 2 x. How would the graph change given the transformations below?  f(x) = 2 x-1  f(x) = 2 x + 4  f(x) = 2 x - 2  f(x) = 2 x + 3

Transformations of Exponential Functions TransformationEquationsDescription Vertical Translation G(x) = b x + c G(x) = b x – c Shifts the graph of f(x) = b x UP c units Shifts the graph of f(x) = b x DOWN c units Horizontal Translation G(x) = b x+c G(x) = b x-c Shifts the graph of f(x) = b x LEFT c units Shifts the graph of f(x) = b x RIGHT c units ReflectionG(x) = -b x G(x) = b -x Reflects the graph f(x) b x about the x-axis Reflects the graph f(x) b x about the y-axis Vertical Shrinking or Stretching G(x) = cb x If c > 1, vertical stretch If 0< c < 1, vertical shrink Horizontal stretching or shrinking G(x) = b cx If c > 1, horizontal shrink If 0< c < 1, horizontal stretch

Example

Logarithmic functions  Definition : Let b > 0 and b not equal to 1. Then y is the logarithm of x to the base be written: y = log b x if and only if b y = x  In other words, a logarithmic graph is the inverse of an exponential.

Graphing logarithms  To graph y = log b x  Rewrite as an exponential equation: b y = x  Make an x/y table, filling in y first.  Graph points.  y = log 3 x xY 0 1 2

Other logarithms Common Logarithm : (base 10) log x = y  10 y = x Natural Logarithm : (base e) log e x = y  ln e = y  e y = x

Properties of y = log b x y = log b x OR x = b y Domain Range Asymptotes (line that graph approaches, but does not touch) Point on all graphs

Determining the Domains of Logarithmic Functions. FYI: the range never changes!  Remember the “argument” must be positive (> 0)  f(x) = log 2 (x – 1)  f(x) = (log 3 x) – 1  f(x) = log 4 |x|  f(x) = log 5 ( x 2 – 4)

Transformations of Logarithmic Functions TransformationEquationsDescription Vertical Translation G(x) = log b x + c G(x) = log b x - c Shifts the graph of f(x)= log b x UP c units Shifts the graph of f(x)= log b x DOWN c units Horizontal Translation G(x) =log b (x + c) G(x) = log b (x - c) Shifts the graph of f(x)= log b x LEFT c units, VERTICAL ASYMPTOTE (x = -c) Shifts the graph of f(x)= log b x RIGHT c units, VERTICAL ASYMPTOTE (x = c) Reflection G(x) = -log b x G(x) = log b (-x) Reflects the graph f(x)= log b x about the x-axis Reflects the graph f(x)= log b x about the y-axis Vertical Shrinking or Stretching G(x) =c log b x If c > 1, vertical stretch If 0< c < 1, vertical shrink Horizontal stretching or shrinking G(x) = log b (cx) If c > 1, horizontal shrink If 0< c < 1, horizontal stretch

Example

Closure