Question

Write a possible exponential function in y = a · bx form that represents each situation described below. Homework Help ✎
Has a y‑intercept of (0, 2) and passes through the point (3, 128).
Passes through the points (0, 4) and (2, 1).​

Answer 1

Answer:

$y=2\cdot4^x\\y=4\cdot\left(\frac{1}{2}\right)^x$

Step-by-step explanation:

We're given both the y-intercept and a point on the graphs of both functions, so our work her is largely just substituting x and y values in the general form of the equation for an exponential function.

For the first one, we can start by using the point (0, 2) to solve for a:

$y=a\cdot b^x\\2=a\cdot b^0\\2=a$

Next, we can use that a-value and the second point to solve for b:

$y=2\cdot b^x\\128=2\cdot b^3\\64=b^3\\4=b$

This gives us the equation $y=2\cdot4^x$ for the first function.

We can repeat the same process for the second function. Solving for a:

$y=a\cdot b^x\\4=a\cdot b^0\\4=a$

And then for b, using the point (2, 1):

$1=4\cdot b^2\\\frac{1}{4}=b^2\\\frac{1}{2}=b$

This gives us the general equation $y=4\cdot\left(\frac{1}{2}\right)^x$