1.3 Rate of Change and Initial Value

Transcription

1 1.3 Rate of Change and Initial Value There are a couple of very important numerical (or quantitative) qualities of functions that we use. We like to know where functions start (initial value) and how steep they go up or down from there (rate of change). Rate of Change The rate of change is a measure of how quickly something changes. If you think of the graph representation of a function, the rate of change would be a measure of how steep the graph is. For many functions, we can t yet nail down the exact rate of change. (That takes a little calculus.) However, we can estimate the rate of change over a given interval (or domain) no matter what form the function takes. Graph form of a function Average rate of change Let s say we wanted to determine an average rate of change over the interval 3, 3 of the function to the left. At the leftmost input of 3 the value of the function is approximately 2.5 while at the rightmost input of 3 the value of the function is approximately 7. To estimate our rate of change, we can use the following formula: This means that in our interval of 3, 3 the average rate of change is about three fourths. This means that it goes up three for every four that moves right. Alternately you can think that the output goes up threefourths of a unit for every increase of one input. Looking at the graph, we see that the average rate of change for this function does not accurately describe the functions behavior, but it does give a sense that the function is moving up. What sort of rate of change would make you think the function was going down? Equation form of a function Let s now determine the average rate of change over the interval 3, 5 for the function 4. To do this, we simply find 3 and 5 and use the above formula, but make sure you know why that formula works Take a look at the graph at the right to see if you think it has an average rate of change of one fourth over the interval 3, 5. What about on the interval 5, 10? What about on the interval 5, 100? What do think will continue to happen to the average rate of change? 21

2 Table and word form of a function Seeing how the equation and graph form of a function work, you should be able to figure out how we could calculate the average rate of change given the table or word form of a function. However, let s look at one example of the table form and find the average rate of change over the interval 2, Using the table you should get an average rate of change of 2. What do you notice about that rate of change as compared to the graph to the right which is a representation of the same function? Initial Value Not only do we want to know how a function behaves in terms of its rate of change, but we also want to know where a function starts initially. That means what is the value of the function at the input of zero. We call this the initial value. Let s look at the graphs of each of the functions we have already examined. What is the value of each function when the input is 0? Initial value of 2 Initial value of 2 Initial value of 1 Graphically it is easy to see the initial value because it is where the graph crosses the axis. We sometimes also refer to this point at the intercept because of this. Finding the initial value is just as straight forward with other function forms. 22

3 Lesson 1.3 Calculate or approximate the average rate of change of the following functions over the interval, and then again over the interval,. Finally, give the initial value

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5 19. A lawyer charges $50 an hour plus a flat attorney s fee of $ A man currently has $10,800 in an interest bearing account. Since the interest is simple and not compound, he has been and will continue to make $100 per year in interest. 21. A Star Wars enthusiast has to be buy the same number of Star Wars toys each year as he already owns total. So if he owns 30 currently, he would buy another 30 bringing his total up to 60. The following year he would have to purchase 60 and so on. He currently has 16 Star Wars toys. 22. What did you notice about the average rate of change over both intervals for problems 5, 11, and 17? What do you think is the reason why that happened? 23. What did you notice about problem number 4 and 16? Why did that happen? Graph both functions and then try to determine why it would happen. 25