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Sec 5.2 : Exponential Functions
1. Exponential Functions :
1.1 Graph of exponential Functions. Function f (x) = ax^ is called exponential function. Here a is a positive number, and x can be any real number (just recall we can calculate rational exponents, then we can get the value of irrational exponents by approximation). Note: why a must be positive? If a is negative, what’s the value of a (^12) =
a? If a
is 0, what the value of a−^1 =
a
If a > 1, then the graph of f (x) = ax^ is increasing. And the left end of the graph tends to the x-axis (or, in other words, the negative x-axis is a horizontal asymptote). If 0 < a < 1, then the graph of f (x) = ax^ is decreasing. And the right end of the graph tends to the x-axis (or, in other words, the positive x-axis is a horizontal asymptote). If a = 1, the graph is simple, because f (x) = 1x^ = 1, f (x) is just a constant function. Ex 1. Below is the graph of f (x) = 2x, how can you get the graph of g(x) = 8x^ and
h(x) = (
)x? (Hint: 8 = 2^3 ,
= 2−^1 )
-4 (^) -2 2 4
5
10
15
20
25
30
Ex 2. How can you get the graph of k(x) = 3 · 2 x−^1?
1.2 Exponential grow/decay and real life examples Exponential model is very common in real life. For example, the compound interest rate of a bank, the splitting of cell, the increase of population, the spread of flu, the nuclear reactions, the decay of Carbon-14, etc.
For example, consider example 6 in your textbook. The increasing of bacteria in the culture is exponential. which means f (x) = P ax, here P is the number of bacteria in the beginning, and x is the time elapsed since the beginning, and a is the exponential increasing rate (which is not given explicitly in the statement). One key point in solving this word problem is to find the beginning. If we set the beginning to be the start of the experiment, we know from the statement that P = 1000, which will dramatically simplify the problem (we just need to find the exponential increasing rate a).
Ex 2. You deposited $1000 into US bank, and the annually compound rate is 3%, how much money will you get in the account after 1 year? after 2 years? after x years?
Ex 3. The case is almost the same as in previous example, the only difference is the interest is compounded quarterly (the annual interest rate is still 3%), how much money do you have in your account after 1 year? after 2 years? after x years? In this example, do you earn more or less interest than in previous example?
Another kind of exponential model is exponential decay. For example, scientists often use the half-life of certain element to determine the age of fossil. For example, if we know the half-life of Carbon-14, and we can infer how much Carbon-14 is in the fossil in the beginning, by measuring how much Carbon-14 is contained in the fossil now, we can approximately know about the age of the fossil. For example, if the fossil contain 1 gram of Carbon-14 in the beginning, and now the fossil contain .25 gram of Carbon-14, and we know the half life of carbon-14 is 5730 years. Then we have:
