The graph of an exponential function passes through the points (
–
1,10) and 1,
2
5
. Write an equation for the function in the form y=a(b)x.
To find the equation of the exponential function, we need to determine the values of a and b in the general form y = a(b)^x.
Given points are (-1, 10) and (1, 25/2).
Using the point (-1, 10):
10 = a(b)^(-1) ... (1)
Using the point (1, 25/2):
25/2 = a(b)^1 ... (2)
25/2 = ab ... (2)
Now, we can solve the system of equations (1) and (2) to find the values of a and b.
From equation (1):
10 = a/b
10b = a ... (3)
Substitute this value of a in equation (2):
25/2 = (10b)b
25/2 = 10b^2
5/2 = b^2
b = √(5/2)
Substitute this value of b in equation (3):
10(√(5/2)) = a
10√(5/2) = a
So, the equation of the exponential function is:
y = 10√(5/2)(√(5/2))^x
or
y = 10√(5/2)(√(5)/√(2))^x
or
y = 10√(5/2)(√(5))^x / (√(2))^x
Therefore, the equation for the function in the form y = a(b)^x is:
y = 10√(5/2)(√(5))^x / (√(2))^x