Please show step, by step how to find the exponential function .?

cidya has done a lot of work, I will just show an easier way.

x −2 −1 0 1 2

f(x)) 1 2 4 8 16

f(x) = a (b^x) ... anything (x in this case) to the 0 power = 1 ... so use that fact to find a immediately

f(0) = a (b^0) = a ... but f(0) is given in the table as 4 ... so a = 4

now use any of the others to find b

f(1) = a (b^1) = 4 (b) ... is given in the table as 8

8 = 4b ... b = 2

f(x) = 4 (2^x)

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now use the rest of the table to check and make sure this is correct

f(-2) = 4 (2^-2) = 4 X 1/2^2 = 4 X 1/4 = 1 so this checks

use the others to also check

f(x) looks exponential. g(x) is not.

https://gyazo.com/44cc448e57d1df463135c1b266c21c34

x y

-2 1

-1 2

0 4

1 8

2 16

y=f(x)

y= a b^x

log y = ln a + x log b

Y = A + B x

Calculate the logarithm of y and keep x as x

y log(y)

1 0

2 0.3010

4 0.6021

8 0.9031

16 1.2041

Do a regression of log(y) on x

Linear Regression

Number of cases 5

∑ X = 0

∑ Y = 3.0103

∑ X^2 = 10

∑ XY = 3.0103

∑ X ∑ Y = 3.0103

b = ( ∑ XY - ∑X ∑Y / n ) / [∑ X^2 - (∑X)^2 / n]

Numerator of b = (3.0103) - (0)(3.0103) / 5 = 3.0103

Denominator of b = [10 - (0)^2 / 5] = 10

b = 3.0103 / 10

Regression coefficient b = 0.30103

Constant = 0.60206

Equation : Y = 0.60206 + 0.30103 x

log a = A

log a = 0.60206

a = 10^0.60206 = 4

log b = B

b = 10^B = 10^0.30103 = 2

The exponential equation is y = a b^x or y = 4 * 2^x

if you substitute x=-2,-1,0,1,2 , you'd get y= 1,2,4,8,16