Find an power function of the form y=ab^x whose graph passes through the points (4,4) and (16,8).
4 = a b^4
8 = a b^16
dividing ... 2 = b^12 ... b = 2^(1/12)
substituting ... 2^2 = a 2^(1/3) ... a = 2^(5/3)
To find a power function of the form y = ab^x that passes through the points (4,4) and (16,8), we can use the following steps:
Step 1: Write the two given points as (x₁, y₁) = (4, 4) and (x₂, y₂) = (16, 8).
Step 2: Substitute the coordinates of the first point into the equation y = ab^x to get the equation 4 = ab^4.
Step 3: Substitute the coordinates of the second point into the equation y = ab^x to get the equation 8 = ab^16.
Step 4: Divide the equation derived from Step 3 by the equation derived from Step 2 to eliminate the variable 'a':
(8 = ab^16) / (4 = ab^4)
2 = b^(16-4)
2 = b^12
Step 5: Take the 12th root of both sides to solve for 'b':
b^(1/12) = 2
Step 6: Simplify the equation by raising both sides to the power of 12:
(b^(1/12))^12 = 2^12
b = 2^12
b = 4096
Step 7: Substitute the value of 'b' back into one of the equations from Step 2 or Step 3 to find 'a'. Let's use Step 2:
4 = a(4096)^4
4 = a * 16777216
a = 4 / 16777216
a = 1/4194304 = 2^-22
Hence, the power function that passes through the points (4,4) and (16,8) is y = (2^-22) * 4096^x.