We are given the exponential function equation:
$y = a \cdot b^x$ and a point on the graph, $(3,4)$. To find $a$ and $b$, follow these steps:

Step 1: Find $a$

Observing the graph, it appears the function passes through $(0,1)$, meaning when $x = 0$, $y = 1$. Using the equation:

$1 = a \cdot b^0$

Since $b^0 = 1$, we get:

$a = 1$

Step 2: Solve for $b$

Substituting $(3,4)$ into the equation:

$4 = 1 \cdot b^3$

$b^3 = 4$

$b = \sqrt[3]{4}$

Step 3: Write the Final Equation

$y = 1 \cdot (\sqrt[3]{4})^x$

or simply:

$y = (\sqrt[3]{4})^x$

Would you like more details or alternative explanations?

Here are some related questions:

1. How do you determine if a function is exponential from a graph?
2. What happens to an exponential function if $a$ is changed?
3. How does the base $b$ affect the growth or decay of the function?
4. Can an exponential function have a negative base? Why or why not?
5. How do you find an exponential function given two points?

Tip: The base $b$ determines the rate of growth. If $b > 1$, the function grows; if $0 < b < 1$, the function decays.