An exponential function f(x)=a(b)^x passes through the points (0,9000) and (2,2250). What are the values of a and b?
To find the values of a and b in the exponential function f(x) = a(b)^x, we can use the given points (0, 9000) and (2, 2250).
Step 1: Using the point (0, 9000)
When x = 0, we have:
f(0) = a(b)^0 = a * 1 = a
Since f(0) is equal to 9000, we can equate it to a:
a = 9000
Step 2: Using the point (2, 2250)
When x = 2, we have:
f(2) = a(b)^2
Substituting the value of a (9000) in the equation, we get:
2250 = 9000(b)^2
Step 3: Solving for b
Divide both sides of the equation by 9000 to isolate (b)^2:
2250/9000 = (b)^2
1/4 = (b)^2
Taking the square root of both sides, we have:
b = ± sqrt(1/4)
b = ± 1/2
Step 4: Determine the correct value of b
Since an exponential function cannot have a negative base, we take the positive value of b:
b = 1/2
Therefore, the values of a and b in the exponential function f(x) = a(b)^x are:
a = 9000
b = 1/2
To find the values of a and b in the exponential function f(x) = a(b)^x, we can use the given points (0,9000) and (2,2250).
Let's start by using the point (0,9000) and substituting the values into the equation:
f(0) = a(b)^0 = 9000
Since any number raised to the power of 0 is 1, the equation simplifies to:
a(1) = 9000
So, we have the equation: a = 9000.
Next, let's use the other given point (2,2250) and substitute the values into the equation:
f(2) = a(b)^2 = 2250
Now we can substitute the value of a (which we found to be 9000) into the equation:
9000(b)^2 = 2250
Divide both sides of the equation by 9000 to isolate (b)^2:
(b)^2 = 2250 / 9000
Simplify:
(b)^2 = 0.25
To solve for b, we take the square root of both sides of the equation:
b = ±√0.25
Now, since b represents the base of an exponential function, it can only be positive. So we take the positive square root:
b = √0.25
Evaluate the square root:
b = 0.5
Therefore, the values of a and b are a = 9000 and b = 0.5, respectively.
since b^0 = 1, you know that a = 9000
so,
y = 9000 b^x
Now use the 2nd point to get b.
b^2 = 9000/2250 = 4
b = 2
y = 9000 * 2^x