SOLUTION: find the formula for an exponential function that passes through the two points given. a) (-1,1/2) and (4,512) b) (-1,7) and (3,4)

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Question 1127693: find the formula for an exponential function that passes through the two points given.
a) (-1,1/2) and (4,512)
b) (-1,7) and (3,4)
Found 3 solutions by greenestamps, MathLover1, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

(NOTE: You can ignore the solution by tutor MathLover1; the question asks for exponential functions -- not linear functions.)

The general form of an exponential function is



Given two points on the graph of an exponential function, the general process for finding the function is

(1) use the coordinates of the two given points in the general form to get two equations in x and y;
(2) divide one equation by the other; that will eliminate a and give you an equation you can solve for b; and
(3) use the value of b in either equation to find the value of a

Your example (a) turns out to have "nice" numbers; so I will demonstrate the process with the second example.

(1)



(2)

to 5 decimal places

(3)



The exponential function is
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!
a) (-1,1/2) and (4,512)
Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (,) and (,)


Start with the slope formula (note: (,) is the first point (,) and (,) is the second point (,))


Plug in ,,, (these are the coordinates of given points)


Subtract (note: if you need help with subtracting or dividing fractions, check out this solver)




Divide the fractions



So the slope is







------------------------------------------------


Now let's use the point-slope formula to find the equation of the line:




------Point-Slope Formula------
where is the slope, and (,) is one of the given points


So lets use the Point-Slope Formula to find the equation of the line


Plug in , , and (these values are given)



Rewrite as



Distribute


Multiply and to get

Add to both sides to isolate y


Combine like terms and to get (note: if you need help with combining fractions, check out this solver)



------------------------------------------------------------------------------------------------------------

Answer:



So the equation of the line which goes through the points (,) and (,) is:


The equation is now in form (which is slope-intercept form) where the slope is and the y-intercept is


Notice if we graph the equation and plot the points (,) and (,), we get this: (note: if you need help with graphing, check out this solver)


Graph of through the points (,) and (,)


Notice how the two points lie on the line. This graphically verifies our answer.




here is better graph:


b) (-1,7) and (3,4)
Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (,) and (,)


Start with the slope formula (note: (,) is the first point (,) and (,) is the second point (,))


Plug in ,,, (these are the coordinates of given points)


Subtract the terms in the numerator to get . Subtract the terms in the denominator to get



So the slope is







------------------------------------------------


Now let's use the point-slope formula to find the equation of the line:




------Point-Slope Formula------
where is the slope, and (,) is one of the given points


So lets use the Point-Slope Formula to find the equation of the line


Plug in , , and (these values are given)



Rewrite as



Distribute


Multiply and to get

Add to both sides to isolate y


Combine like terms and to get (note: if you need help with combining fractions, check out this solver)



------------------------------------------------------------------------------------------------------------

Answer:



So the equation of the line which goes through the points (,) and (,) is:


The equation is now in form (which is slope-intercept form) where the slope is and the y-intercept is


Notice if we graph the equation and plot the points (,) and (,), we get this: (note: if you need help with graphing, check out this solver)


Graph of through the points (,) and (,)


Notice how the two points lie on the line. This graphically verifies our answer.
Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
.

            Below find my solution for case a).

find the formula for an exponential function that passes through the two points given.
a) (-1,1/2) and (4,512) 

General exponential function formula


    y = .        (1)


At  x= -1,  y=   formula (1) becomes    =     (2).


At  x= 4,  y= 512  formula (1) becomes   512 =     (3).


Divide  (3)  by (2).  You will get


     = ,   or, equivalently,

    1024 = ,

     = .


It implies  b = 4,  so half of the problem is just solved.


Now substitute the found value b= 4 into equation (3). You will get

  
    512 = ,  

    
which implies  a =  =  = 2.


Answer.  The function is  y = .

Solved.


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