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33 Cards in this Set

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What are the two equations to represent an exponential function?
y=ab^t

y=ae^(kt)
while linear functions change at a certain rate, exponential functions change at a certain...
percentage rate
What do a and b stand for in the equation y=ab^t?
a= initial value at t=0
b=growth factor=1+/-r
what is r and how is it related to b in y=ab^t?
r=the decimal representation of the percent rate of change. It is referred to as the annual or yearly percent rate.

b=1+/-r
If Q=ab^t then for

1. 0<b<1 the function is.....

2. b>1 the function is....
decreasing

increasing
For a table of data of f(x), where the change in x is constant,

Then if the difference of consectutive y-vals is constant the table represents a .....
linear function
To check if a table represents an exponential function one should check that
ratio of consecutive y values are constant. If the ratios are constant then the table is an exponential function
Let (1,3) and (4,1) be points of an exponential function. What are the steps to find an equation for this function?
1. let y1=ab^t and y2=ab^t
2. plug in points into each equation:
3=ab^1 and 1=ab^4
3. Take the ratio of y1/y2
i.e. (ab^1)/(ab^4)=(3/1)
4. cancel the a's and solve for b:
b^1/b^4 = b^3=(3/1)
b=3^(1/3)
5. Plug in b to either y1 or y2 and use a point to solve for a
True or false, exponential growth will always outpace linear growth in the long run
TRUE!
the graph of the exponential function intersects the y axis at....
a in ab^t
The larger b in ab^t, the more the graph tends away or toward the y axis?
toward
A horizontal asymptote is given by the equation...
y=k for k a constant
as x gets closer to ________ and f(x) remains close to some line y=k, then the graph of f(x) has a __________ asymptote at __________
positive or negative infinity, horizontal,
the line y=k
True or false, a graph can have a horizontal asymptote if as x gets closer to NEGATIVE infinity, f(x) stays close to some line y=k
true
k in Q=ae^(kt) is called the...
continuous growth rate
How are the equations y=ab^t and y=ae^(kt) related?
if you let b=e^k, then they are the same equation
For k, the continuous growth rate, if k>0 then Q=ae^(kt) is increasing or decreasing?
increasing.
If k<0 then Q would be decreasing
Which graph would most likely resemble the graph of y=ae^t:

y=a(4)^t
y=a(.01)^t
y=a(2)^t
since e=2.71828, the graph would look like y=a(2)^t
If a problem tells you there is a bank account paying 12% annual interest, what formula will you use to represent the balance of the bank account? Why?
use balance=ab^t because 0.12 is equal to r and b=1+/- r

You know .12 is equal to r because the problem says it is an ANNUAL as opposed to continuous interest.
If a question asks you to write an equation for an exponential function that has a continuous growth rate of 8%, what formula will you use and why?
Use y=ae^(kt) because the CONTINUOUS growth rate =k so k=0.08
If you have an exponential equation in the form of y=ab^t, how would you obtain the continuous rate, k?
ln(b)=k
If you have an exponential equation in the form of y=ae^(kt), how would you obtain the annual rate, r?
(e^k)-1=r
If a problem asks you to solve for the doubling time, it is asking you to solve for _________ by setting the equation equal to twice its ____________
time, initial value.
True or false, if a town has a population of 3000 at t=0 and the population grows by 6% per year, we are being given k, and should thus use the equation y=ae^(kt)
False, we are given r so use ab^t since b=1+/- r
What kind of function would a statement "a town grows by 200 people per year" represent?
linear
What kind of function would a statement "a town shrinks by 4% each year" represent?
exponential
While linear functions deal with constant rates of change, exponential functions deal with constant rates of .......
percent change
What steps would you take to find the half life of a substance that is continuously decaying at a rate of 3%?
1. let y=ae^(kt)
2. k=-0.03 (negative since decay)
so y=ae^(-0.03t)
3. 0.5a=ae^(-0.03t)
4. 0.5=e^(-0.03t)
5 take ln of both sides and solve for t
(a^m)*(a^n)=?
a^(m+n)
(a^m)/(a^n)=?
a^(m-n)
True or false:

(a + b)^n = a^n + b^n
false
(b*a^m)^n=?
(b^n)*(a^mn)
The independent variable in an exponential function is always found where?
In the exponent