8-1 Exploring Exponential Models

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1 8- Eploring Eponential Models Eponential Function A function with the general form, where is a real number, a 0, b > 0 and b. Eample: y = 4() Growth Factor When b >, b is the growth factor Eample: y = (3) b = 3 which is greater than so it is the factor and the function is one of growth. Decay Factor When b <, b is the decay factor Eample: y = (¼) b = ¼ which is less than so it is the factor and the function is one of decay. Asymptote - A line that a graph approaches as increases in absolute value. Graphing Eample: Graph y =. Step Make a table. Step Graph the coordinates. Connect the points with a smooth curve y

2 Percent Increase or Decrease The growth factor, b, can be represented as b =, where r is the rate of increase. The decay factor, b, can be represented as b =, where r is the rate of decrease. eamples: Find the percent increase or decrease. ) y = (.3) b =.3 which is > so it is an increase (ep. growth). so b = + r.3 = + r substituting.3 for b 0.3 = r subtracting from both sides So the percent of increase is % ) y = 0.35(0.65) b = 0.65 which is < so it is a decrease (ep. decay). so b = - r 0.65 = - r substituting 0.65 for b = -r subtracting from both sides 0.35 = r multiplying both sides by - So the percent of decrease is % 8- Properties of Eponential Functions The function f() = b is the parent of a family of eponential functions for each value of b. The factor a in y = ab stretches, shrinks, and/or reflects the parent. Summary Families of Eponential Functions Parent functions (b > 0, b = ): Stretch ( a > ) Shrink (0 < a < ) } Reflection (a < 0) in -ais Translation (horizontal by h; vertical by k): Combined:

3 eample : Graphing y = ab for 0 < a < Graph y = and y =. Label the asymptote of each graph. Step Make a table Step Graph the function y y y The y-intercept is The asymptote is The y-intercept is eample : Translating y = ab Graph the stretch y = 8( ) and then the translation y = 8( ) Step Graph y = 8( ). The horizontal asymptote is Step For y = 8(½) + + 3, h = - and k = 3. So shift the y = 8(½) graph units and units. The horizontal asymptote is y

4 Base e Eponential functions with a base of e are useful for describing growth or decay. Your calculator has a key for e. eamples: ) e 4 ) e -3 3) Radioactive Decay Use the standard equation where e y = the amount of the radioactive substance a = the initial amount of the substance b = the decay factor = the time Compound Interest y ab rt Use the standard equation y ab as A Pe where A = the final Amount P = the Principal investment (initial amount) e = e r = the Rate of interest t = time 8-3 Logarithmic Functions as Inverses

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6 A logarithmic function is the of an eponential function. You can graph y = log b as the inverse of y =. Families of Logarithmic Functions Parent functions (b > 0, b ) Stretch ( a > ) Shrink (0 < a < ) } Reflection (a < 0) in -ais Translation (horizontal by h; vertical by k) Combined The function y = log b is the inverse of y =. Since (0,) and (,b) are points on the graph of y = b, (, ) and (, ) are points on the graph of y = log b. Since the -ais is the asymptote for y = b, the -ais is the asymptote for y = log b.

7 You can graph y = log b ( - h) + k by translating the graph of y = log b horizontally by and vertically by. 8-4 Properties of Logs

8 You can write the sum or difference of logarithms with the same base as logarithm

9 You can sometimes write a single logarithm as a sum or difference of logarithms 8-5 Eponential and Logarithmic Equations An equation of the form b c = a, where the eponent includes a variable, is an. You can solve an eponential equation by taking the of each side of the equation.

10 An equation that includes a logarithmic epression is called a

11 8-6 Natural Logarithms Base e e is an irrational number approimately equal to Eponential functions with a base of e are useful for describing continuous, y = e The function y = e has an inverse, the function. The properties of common logarithms apply to natural logarithms also. You can use the properties of logarithms to solve natural logarithmic equations.

12 You can use natural logarithms to solve eponential equations.