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)

Algebra ->  Equations -> 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)      Log On


   

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) About Me  (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

y+=+ab%5Ex

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)
ab%5E%28-1%29+=+7
ab%5E3+=+4

(2)
b%5E4+=+4%2F7
b+=+%284%2F7%29%5E%281%2F4%29+=+.86944 to 5 decimal places

(3)
a%28.86944%29%5E%28-1%29+=+7
a+=+7%28.86944%29+=+6.08608

The exponential function is

y+=+6.08608%280.86944%29%5Ex

Answer by MathLover1(20850) About Me  (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 (-1,1%2F2) and (4,512)


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: (x%5B1%5D,y%5B1%5D) is the first point (-1,1%2F2) and (x%5B2%5D,y%5B2%5D) is the second point (4,512))


m=%28512-1%2F2%29%2F%284--1%29 Plug in y%5B2%5D=512,y%5B1%5D=1%2F2,x%5B2%5D=4,x%5B1%5D=-1 (these are the coordinates of given points)


m=+%281023%2F2%29%2F%285%2F1%29 Subtract (note: if you need help with subtracting or dividing fractions, check out this solver)




m=1023%2F10 Divide the fractions



So the slope is

m=1023%2F10





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




------Point-Slope Formula------
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and (x%5B1%5D,y%5B1%5D) is one of the given points


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


y-1%2F2=%281023%2F10%29%28x--1%29 Plug in m=1023%2F10, x%5B1%5D=-1, and y%5B1%5D=1%2F2 (these values are given)



y-1%2F2=%281023%2F10%29%28x%2B1%29 Rewrite x--1 as x%2B1



y-1%2F2=%281023%2F10%29x%2B%281023%2F10%29%281%29 Distribute 1023%2F10


y-1%2F2=%281023%2F10%29x%2B1023%2F10 Multiply 1023%2F10 and 1 to get 1023%2F10

y=%281023%2F10%29x%2B1023%2F10%2B1%2F2 Add 1%2F2 to both sides to isolate y


y=%281023%2F10%29x%2B514%2F5 Combine like terms 1023%2F10 and 1%2F2 to get 514%2F5 (note: if you need help with combining fractions, check out this solver)



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Answer:



So the equation of the line which goes through the points (-1,1%2F2) and (4,512) is:y=%281023%2F10%29x%2B514%2F5


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=1023%2F10 and the y-intercept is b=514%2F5


Notice if we graph the equation y=%281023%2F10%29x%2B514%2F5 and plot the points (-1,1%2F2) and (4,512), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=%281023%2F10%29x%2B514%2F5 through the points (-1,1%2F2) and (4,512)


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 (-1,7) and (3,4)


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: (x%5B1%5D,y%5B1%5D) is the first point (-1,7) and (x%5B2%5D,y%5B2%5D) is the second point (3,4))


m=%284-7%29%2F%283--1%29 Plug in y%5B2%5D=4,y%5B1%5D=7,x%5B2%5D=3,x%5B1%5D=-1 (these are the coordinates of given points)


m=+-3%2F4 Subtract the terms in the numerator 4-7 to get -3. Subtract the terms in the denominator 3--1 to get 4



So the slope is

m=-3%2F4





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




------Point-Slope Formula------
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and (x%5B1%5D,y%5B1%5D) is one of the given points


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


y-7=%28-3%2F4%29%28x--1%29 Plug in m=-3%2F4, x%5B1%5D=-1, and y%5B1%5D=7 (these values are given)



y-7=%28-3%2F4%29%28x%2B1%29 Rewrite x--1 as x%2B1



y-7=%28-3%2F4%29x%2B%28-3%2F4%29%281%29 Distribute -3%2F4


y-7=%28-3%2F4%29x-3%2F4 Multiply -3%2F4 and 1 to get -3%2F4

y=%28-3%2F4%29x-3%2F4%2B7 Add 7 to both sides to isolate y


y=%28-3%2F4%29x%2B25%2F4 Combine like terms -3%2F4 and 7 to get 25%2F4 (note: if you need help with combining fractions, check out this solver)



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Answer:



So the equation of the line which goes through the points (-1,7) and (3,4) is:y=%28-3%2F4%29x%2B25%2F4


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=-3%2F4 and the y-intercept is b=25%2F4


Notice if we graph the equation y=%28-3%2F4%29x%2B25%2F4 and plot the points (-1,7) and (3,4), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=%28-3%2F4%29x%2B25%2F4 through the points (-1,7) and (3,4)


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

Answer by ikleyn(52788) About Me  (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 = ab%5Ex.        (1)


At  x= -1,  y= 1%2F2  formula (1) becomes   1%2F2 = a%2Ab%5E%28-1%29    (2).


At  x= 4,  y= 512  formula (1) becomes   512 = a%2Ab%5E4    (3).


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


    512%2F%28%281%2F2%29%29 = b%5E4%2Fb%5E%28-1%29,   or, equivalently,

    1024 = b%5E5,

    4%5E5 = b%5E5.


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 = a%2A4%5E4,  

    
which implies  a = 512%2F4%5E4 = 512%2F256 = 2.


Answer.  The function is  y = 2%2A4%5Ex.

Solved.