Find a possible formula for the exponential function?
1.(0,40),(5,15)
2.(-2,12/9),(2,27/4)
I'll do the easy one, then you show me how you did the easy one:
(-2,12/9),(2,27/4)
let the function be y = a(b)^x
case 1: 12/9 = a(b)^-2
case 2: 27/4 = a(b)^2
divide the second by the first:
[a(b)^2] / [a(b)^-2] = (27/4) / (12/9)
b^4 = 81/16
b = ± 3/2
sub back into case 2 equation:
if b = 3/2
27/4 = a(3/2)^2
27/4 = (9/4)a
a = 27/9= 3
if b = -3/2
27/4 = a(-3/2)^2
27/4 =(9/4)a
a = 3
y = 3(±3/2)^x
check: if x = 2
y = 3(9/4) = 27/4
if x = -2
y = 3(4/9) = 12/9 , my answer is correct
To find a possible formula for an exponential function, we can use the general form of an exponential function:
y = a * b^x,
where 'a' is the initial value or y-intercept, 'b' is the base, and 'x' is the exponent.
Let's solve for the formula using the given points:
1. (0, 40) and (5, 15):
Substituting the first point (0, 40) into the formula, we get:
40 = a * b^0
40 = a * 1
40 = a
Now, substituting the second point (5, 15) into the formula, we get:
15 = 40 * b^5
Dividing both sides by 40:
15/40 = b^5
3/8 = b^5
To find the value of 'b', we can take the fifth root of both sides:
(b^5)^(1/5) = (3/8)^(1/5)
b = 3/8
Therefore, the possible formula for the exponential function is:
y = 40 * (3/8)^x.
2. (-2, 12/9) and (2, 27/4):
Similarly, substituting the first point (-2, 12/9) into the formula, we get:
12/9 = a * b^(-2)
Now, substituting the second point (2, 27/4) into the formula, we get:
27/4 = a * b^2
Dividing both sides by (12/9):
(27/4) / (12/9) = (a * b^2) / (a * b^(-2))
27/4 * 9/12 = b^2 / b^(-2)
Simplifying the equation:
27/4 * 9/12 = b^(2 - (-2))
243/16 = b^4
To find the value of 'b', we can take the fourth root of both sides:
(b^4)^(1/4) = (243/16)^(1/4)
b = 3/2
Therefore, the possible formula for the exponential function is:
y = (12/9) * (3/2)^x.