Which exponential function passes through the points (1, 16) and (4, 128)?

To find the exponential function that passes through the points (1, 16) and (4, 128), we can start by assuming a general form of the exponential function, which is:

y = a * b^x

Here, a is the initial value (the value of y when x = 0), and b is the base of the exponential. We need to find the values of a and b that fit the given points.

We can plug the first point (1, 16) into the equation:

When x = 1, y = 16

16 = a * b¹

Thus, we get:

a * b = 16 (Equation 1)

Next, we use the second point (4, 128):

When x = 4, y = 128

128 = a * b⁴

This gives us:

a * b⁴ = 128 (Equation 2)

Now, we have a system of two equations:

• Equation 1: a * b = 16
• Equation 2: a * b⁴ = 128

We can solve for a using Equation 1:

a = 16 / b

Now, substitute this value of a into Equation 2:

(16 / b) * b⁴ = 128

Simplifying the equation:

16 * b³ = 128

Divide both sides by 16:

b³ = 8

Taking the cube root of both sides gives:

b = 2

Now that we have the value of b, we can substitute it back into Equation 1 to find a:

a * 2 = 16

Solving for a gives:

a = 8

Now we have a and b:

a = 8

b = 2

Putting it all together, the exponential function that goes through the points (1, 16) and (4, 128) is:

y = 8 * 2^x

This equation expresses the relationship perfectly, showing how the function grows exponentially as x increases.