Please show step, by step how to find the exponential function .
The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Give the exponential models for those that are. HINT [See Example 1.] (If an answer does not exist, enter DNE.)
x −2 −1 0 1 2
f(x)) 1 2 4 8 16
g(x) 9 0 1 0 9
f(x) =
g(x) =
david
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)
=================
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
cidyah
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