Sections 9.6 + 9.7Power, Exponential, Log, and

Polynomial Functions

• In computer science, the number of operations required for a program to solve a problem is often stated as a function of the size of the input data set.

• For example, program A may be able to complete the job in 5n4 steps while program B might take 1.4n steps.

• Which is better for a small set of data?

• Which is better for a large set of data?

• Use the following two functions to complete the table

xxgxxf )4.1()(5)( 4 x 1 2 10 30 60 100

g(x)

f(x)

•Which function is growing faster?

•Where do the two functions intersect?

•Use your calculator to find a window where you can see their intersection

• Now use the following functions to complete the table

• Which function is growing faster?

• Where do they intersect?– Use your graphing calculator to find out.

x 1 2 10 30 60 100 2000 3000

h(x)

k(x)

)ln(3)(2)( 31

xxkxxh

• We have now encountered three basic families of functions– Linear– Power– Exponential

• We can find a unique function for each given two points

• Let’s find one of each that go through the points (-1, ¾) and (2, 48)

• Let’s take a look at their graphs– Use a window of -3 ≤ x ≤ 3 and -10 ≤ y ≤ 50

bmxxf )(nkxxg )(xabxh )(

Modeling Data• Think way back to chapter 3 we used

exponential functions to model quantities that were both growing and decaying

• Why would we like to be able to find a function that models a given data set?

• The following table contains the population of the Houston-Galveston-Brazonia metro area

• Create a scatter plot of the data (use t = 0 to represent the year 1900)– What type of shape does the data have?– Use your calculator to fit an exponential function

to the data– Use your calculator to fit a power function to the

data

Year 1940 1950 1960 1970 1980 1990 2000

Population (thousands)

737 1070 1583 2183 3122 3733 4672

• Graph your two functions together with your scatter plot

• Use each model to predict the population in 1975 and 2010

• What do you think about your answers?

• What do you think about predicting the population in 2050?