Question 982411: An exponential function has a y-intercept of 31 and a horizontal asymptote of -73. In addition to this information, the slope of the line passing between any two points on the graph is positive. Determine the equation of any function that satisfies the above information.
WHAT I KNOW: given y-intercept= 31
for y intercept, x=0
y intercept = (0,31)
I plotted this on a graph I made and now i am just confused as to what step to take next in order to find the equation.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! There is some room for confusion, just as there is room for multiple answers.
A popular exponential function is   ,
but there is nothing sacred to that number  until you get to calculus.
The function   is also an exponential function.
A power of any base is an exponential function, but if you try a base such as  , you get a reversed graph:  .
We can use any base greater than  and we would get an exponential function with the same basic qualities:
1) it has a y- intercept of , so it passes through the point (0,1),
2) its horizontal asymptote is y=0, and
3) the slope of the line passing between any two points on the graph is positive.
So far, quality number 3) meets the requisites of the problem.
To nail 1) and 2) we can stretch the graph vertically, and translate it down.
The functions above had an (invisible) as a coefficient multiplying the power,
and the difference between the y-value of the asymptote and the y-intercept was .
If we multiply times  , as in  ,
the difference between the y-value of the asymptote and the y-intercept will be  ,
which happens to be the difference between "a y-intercept of 31 and a horizontal asymptote of -73".
If next we move the graph down by  units by adding a  ,
we get  :  .
|
|
|
