Find an power function of the form y=ab^x whose graph passes through the points (4,4) and (16,8).
Hey, you do some.
To find a power function of the form y = ab^x that passes through the points (4,4) and (16,8), we can start by setting up a system of equations using the given points.
Let's consider the point (4,4). With x = 4 and y = 4, we have the equation:
4 = ab^4 -- Equation 1
Similarly, using the other point (16,8), with x = 16 and y = 8, we have:
8 = ab^16 -- Equation 2
We now have a system of two equations with two unknowns (a and b). We need to solve this system to find the values of a and b.
To eliminate the variable 'a' from the equations, we can divide Equation 2 by Equation 1:
(8/4) = ab^16 / ab^4
2 = b^12
To solve for b, we take the twelfth root of both sides:
b = ∛2
Now we substitute the value of b back into Equation 1 to find a:
4 = a (∛2)^4
4 = a * 2
a = 2
So the values of a and b are a = 2 and b = ∛2.
Finally, we can write the power function in the form y = ab^x using these values:
y = 2(∛2)^x
Therefore, the power function that passes through the points (4,4) and (16,8) is y = 2(∛2)^x.