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.
y=
To find the equation of the exponential function in the form y=a(b)^x, we need to find the values of a and b.
Let's substitute the given points into the equation:
When x = -1, y = 10
10 = a(b)^(-1)
When x = 1, y = 25/2
25/2 = a(b)^1
We have a system of equations:
a(b)^(-1) = 10
a(b)^1 = 25/2
To eliminate a, let's divide the second equation by the first equation:
(a(b)^1) / (a(b)^(-1)) = (25/2) / 10
a(b)^2 = 5/2
Simplify the equation:
(b)^2 = (5/2) / a
(b)^2 = (5/2) / (a/1) (To convert 'a' as a fraction, write a/1)
(b)^2 = (5/2) * (1/a) (To divide fractions, multiply the first fraction by the reciprocal of the second fraction)
(b)^2 = 5 / (2*a)
Now substitute the value of (b)^2 into the second equation:
5 / (2*a) = 25/2
10*a = 5
Solve for a:
a = 5/10
a = 1/2
Now substitute the value of a into the first equation to solve for b:
(1/2)(b)^(-1) = 10
b^(-1) = 20
1/b = 20
b = 1/20
Now that we have the values of a and b, we can write the equation:
y = (1/2)(1/20)^x