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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.

Find the equation of the exponential function given the graph below: y = a * b^x

Learn how to determine the equation of an exponential function given a graph with a point (3,4). This step-by-step solution explains how to find 'a' and 'b' using algebraic methods, making it suitable for students in Grades 9-11.

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