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.
Step-by-step 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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<h3>1.</h3>
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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<h3>2.</h3>
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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<h3>3.</h3>
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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<em>Additional comment</em>
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 <em>natural numbers</em>, 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.
