Chapter 3- exponential functions
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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%?
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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