Find the equation for the exponential function that passes through the pairs of points given. Through (-1,2) and (3,1).
I don't know if this is the right way to work this or what the next step is i am just very confused. Please help, thank you!
y=Ab^x
2=Ab^1
1=Ab^3
2=(1/b^3)=b^-3
2=b^-3(b^-3)
2=b^-6
Now
Ahem
2 = Ab^-1 = A/b, so A = 2b
1 = Ab^3 = 2b^4
b = (1/2)^1/4
2 = A*2^(1/4)
A = 2^(3/4)
y = 2^(3/4) (1/2)^(x/4)
or
y = ∜(8/2^x)
To find the equation for the exponential function that passes through the given pairs of points (-1,2) and (3,1), you need to solve the system of equations formed by substituting these points into the general form of an exponential function, y = Ab^x.
Start by substituting the coordinates of the first point (-1,2) into the equation:
2 = Ab^(-1)
Next, substitute the coordinates of the second point (3,1) into the equation:
1 = Ab^3
Now, you have two equations with two unknowns (A and b). To eliminate A and solve for b, divide the two equations:
(2 / 1) = Ab^(-1) / Ab^3
2 = b^(-1 - 3)
2 = b^(-4)
b^4 = 2
Now, take the fourth root of both sides to solve for b:
b = ∛(2)
Next, substitute this value of b into one of the original equations to solve for A. Let's use the first equation:
2 = A(∛2)^(-1)
2 = A / (∛2)
2 * (∛2) = A
2∛2 = A
So, the equation for the exponential function that passes through the points (-1,2) and (3,1) is:
y = 2∛2 * (∛2)^x
Simplifying this equation further, we can also write it as:
y = 2∛2 * 2^(x/3)
Therefore, the exponential function that passes through the given points is y = 2∛2 * 2^(x/3).