To find the formula for the exponential function that passes through the points $(0, 4)$ and $(2, 144)$, we can assume that the function has the form:

$y = a \cdot b^x$

where $a$ and $b$ are constants. We will use the two points provided to solve for $a$ and $b$.

Step 1: Use the Point $(0, 4)$

When $x = 0$, $y = 4$: $4 = a \cdot b^0$ Since $b^0 = 1$, we get: $a = 4$

So, the equation becomes: $y = 4 \cdot b^x$

Step 2: Use the Point $(2, 144)$

When $x = 2$, $y = 144$: $144 = 4 \cdot b^2$

Divide both sides by 4: $36 = b^2$

Take the square root of both sides to solve for $b$: $b = 6$

Final Equation

Now we can substitute $a = 4$ and $b = 6$ back into the equation: $y = 4 \cdot 6^x$

So, the exponential function that goes through the points $(0, 4)$ and $(2, 144)$ is: $y = 4 \cdot 6^x$

Would you like more details, or do you have any questions?

Here are some related questions for further exploration:

1. How do you determine if a function is exponential from a graph alone?
2. What are other common forms of exponential functions?
3. How would the equation change if $a$ was negative?
4. How can you verify this function using additional points on the graph?
5. What happens to the graph if the base $b$ is less than 1?

Tip: To find an exponential equation given two points, first use the $y$-intercept (where $x = 0$) to solve for the initial value $a$, then use a second point to solve for the base $b$.