To find the exponential function of the form f(x) = ab^x that passes through the given points, we need to use the y-intercept and the point we know. Here are the step-by-step instructions:
- Start with the general form of an exponential function: f(x) = ab^x.
- Given the y-intercept (0, 2), substitute x = 0 into the equation:
- Now we have our function partially defined as f(x) = 2b^x.
- Next, we use the second point (1, 25). Substitute x = 1 into the equation:
- This simplifies to:
- Now divide both sides by 2:
- We now have the values for a and b, so the full exponential function can be written as:
- To confirm, you can check that it passes through both points:
- At (0, 2): f(0) = 2(12.5)^0 = 2, which matches the y-intercept.
- At (1, 25): f(1) = 2(12.5)^1 = 25, which matches the second point.
f(0) = ab^0 implies f(0) = a = 2.
f(1) = 2b^1 = 25.
2b = 25.
b = 25 / 2 = 12.5.
f(x) = 2(12.5)^x.
Thus, the exponential function that meets both criteria is f(x) = 2(12.5)^x.