Section 9.1 Power Functions

It turns out the Pi, or π, is actually a constant of proportionality Let’s talk about Pi Not pie, but Pi What is it? Where does it come from? How do we use it? It turns out the Pi, or π, is actually a constant of proportionality C = πd A = πr2 It is the ratio of the circumference of a circle to its diameter It is the ration of the area of a circle to the square of its radius

A quantity y is directly proportional to a power of x if A quantity y is inversely proportional to a power of x if Functions of this form are called power functions

Which of the following are power functions and identify the k and the n (recall )

Now let’s look at power functions with various exponents Let k = 1 for the following functions n = 0, 1, 2, 3, -2, -3, 1/2, and 1/3 In each case Look at the graph of the function, Note the general shape, Note the concavity, Note the x-intercepts, Note the asymptotes You may want to make a chart and write down what each function does

The graphs from the preceding slide are the 6 basic graphs we can compare all graphs two If we use k = -1, we get 6 more basic graphs which are reflected over the x-axis, but have the same characteristics as the ones we just looked at For example, which of our 6 functions would we compare the following to?

Example On a map 1/4 inch represents about 1 mile. How can we represent this relationship as a power function? What is the constant of proportionality? What is the exponent? How far apart are two locations if they are 2.4 inches apart on the map? In your groups try problems 15, 26, and 32