Answer:
- a. decay; b. growth; c. decay; d. neither
- r = 4; a = 1; y = 1·4^x
- a. an = 3(5^(n-1)); b. f(x) = (3/5)(5^x); c. exponential growth; d. y-intercept: 3/5; first term: 3.
Explanation:
There are two kinds of exponential problems here.
- exponential functions of the form f(x) = a·b^x
- exponential sequences of the explicit form an = a1·r^(n-1)
The second problem gives you a table that suggests the sequence form, but it asks for the exponential function form. The third problem does something similar.
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1.
In an exponential function of the form f(x) = a·b^x, the function grows if b>1 and decays if b<1. Using this check, we can easily answer ...
a. 0.4 < 1 . . . decay
b. 1.3 > 1 . . . growth
c. 1/2 < 1 . . . decay
d. 1 = 1 . . . neither growth nor decay; the function is constant: j(x) = 1.
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2.
The value of x is given starting at 1, so we can consider this a geometric sequence. The common ratio is r = 16/4 = 4. The first term is a1 = 4, so the explicit formula for the sequence is ...
an = 4·4^(n-1)
When this is expanded to get rid of the constant in the exponent, we have ...
an = 4·(4^n)·(4^-1) = 1·4^n
We recognize this form as matching the functional form f(x) = a·r^x. The multiplier of the exponential factor is a=1. In summary, ...
r = 4; a = 1; f(x) = 1·4^x
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3.
The first term of this geometric sequence is a1 = 3. The common ratio is r = 15/3 = 5. Using the explicit formula, we have ...
a. explicit form: an = 3·5^(n-1)
Using the method of question 2 to write the functional form, we find ...
an = 3(5^n)(5^-1) = (3/5)(5^n)
b. functional form: f(x) = (3/5)(5^x)
c. function family: exponential growth functions
d. y-intercept: (3/5) . . . . read this from the f(x) form
1st term: the first term listed in the given sequence is 3
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Additional comment
The "y-intercept" of a sequence is irrelevant (undefined), as the sequence term numbering starts with 1, not 0. The domain of the explicit formula is natural numbers, which does not include 0.
Similarly, the "first term" of a function f(x) needs further definition. Here, we've answered the question by saying the first term is f(1). There is no conventional definition of a "first term" for a continuous function.