Question

Use substitution to find an exponential function, y = a * b^x, that passes through the points (2, 16) and (5, 1.024).

a) y = 4 * 2^x
b) y = 2 * 4^x
c) y = 8 * 2^x
d) y = 2 * 8^x

Answer 1

To determine the exponential function that passes through the given points, we first substituted them into the general function and found the value of 'a' using the first point. Next, we substituted the second point and solved for 'b'. The solving process indicates that the correct exponential function is y = 4 * 2^x, which is option a).

Step-by-step explanation:

To find an exponential function that passes through the points (2, 16) and (5, 1.024), we can substitute these points into the general form of the exponential function, y = a * b^x, and solve for the constants a and b.

Let's start with the first point (2, 16).

1. For x = 2, y should be 16, so we can write the equation 16 = a * b^2.
2. Solving this equation for a gives us a = 16 / b^2.

Next, we turn to the second point (5, 1.024).

1. Substitute x = 5, y = 1.024, and a = 16 / b^2 into the original equation to get 1.024 = (16 / b^2) * b^5.
2. Now, simplify and solve for b, which results in b = 2.
3. Return to the found value of a in step 2 to get a = 16 / 2^2 = 4.

So, the exponential function is y = 4 * 2^x, which is option a).