SOLUTION: How do I find the formula for an exponential function that passes through the two given points? (0,6) and (3,750)

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Question 1137506: How do I find the formula for an exponential function that passes through the two given points?
(0,6) and (3,750)
Found 3 solutions by MathLover1, jim_thompson5910, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
original exponential formula is
y+=+ab%5Ex
use given points (0,6) and (3,750), set up system and solve for a and b

y+=+ab%5Ex .....if (0,6)

6+=+ab%5E0
6+=+a%2A1
a=6+

y+=+6b%5Ex.....if (3,750)
750+=+6b%5E3
b%5E3=750%2F6
b%5E3=125
b%5E3=5%5E3
b=5

and, your exponential equation is y+=+6%2A5%5Ex






Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

The general template for an exponential function is
y+=+a%2Ab%5Ex
where 'a' and 'b' are fixed values (constants).

The first point given to us is (0,6) meaning that x = 0 and y = 6 pair up together. Let's plug them both into the template to see what happens
y+=+a%2Ab%5Ex

6+=+a%2Ab%5E0 replace x with 0; replace y with 6

6+=+a%2A1 anything to the zeroth power is equal to 1

6+=+a one times any number is itself

a+=+6 So we have one of the constants we need.

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Now plug in the other point (x,y) = (3,750). Also we will plug in a = 6 that we found earlier. Then solve for b

y+=+a%2Ab%5Ex

y+=+6%2Ab%5Ex replace 'a' with 6

750+=+6%2Ab%5E3 plug in (x,y) = (3,750)

750%2F6+=+%286%2Ab%5E3%29%2F6 Divide both sides by 6

125+=+b%5E3

b%5E3+=+125

root%283%2Cb%5E3%29+=+root%283%2C125%29 Apply the cube root to both sides. You can also think of it as raising both sides to the 1/3 power.

b+=+5 We found the other constant

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Therefore,
a = 6
b = 5

Making the function be f%28x%29+=+6%285%29%5Ex which is the same as y+=+6%285%29%5Ex since y = f(x).

As a check, plug in x = 0 to get
f%28x%29+=+6%285%29%5Ex
f%280%29+=+6%285%29%5E0
f%280%29+=+6%281%29
f%280%29+=+6
So we get the proper output for the input x = 0. Repeat for the input x = 3 as well
f%28x%29+=+6%285%29%5Ex
f%283%29+=+6%285%29%5E3
f%283%29+=+6%28125%29
f%283%29+=+750
that output is correct as well. The function is confirmed. Both points are on the graph of this function curve.

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The general form of an exponential function is

y+=+ab%5Ex

Make two equations in a and b using the x and y values of the two given points:

(0,6) --> 6+=+ab%5E0
(3,750) --> 750+=+ab%5E3

Solving the first equation gives you the value of a; then plugging that value for a in the second equation gives you an equation you can solve for b.