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Find an equation for the exponential function graphed.



The graph shows an exponential function \( f(x) \), and it passes through the point \( (0, \sqrt{2}) \). Based on this, we can derive the general form of the equation of the function.

### General form of an exponential function:
The general form of an exponential function is:

\[
f(x) = ab^x
\]

Where:
- \( a \) is the initial value (the \( y \)-intercept, when \( x = 0 \)),
- \( b \) is the base of the exponential function.

### Step 1: Use the given point \( (0, \sqrt{2}) \)
From the graph, we know the function passes through \( (0, \sqrt{2}) \). This tells us that when \( x = 0 \), \( f(x) = \sqrt{2} \). Plugging this into the general equation:

\[
f(0) = a \cdot b^0 = a \cdot 1 = a
\]
So, \( a = \sqrt{2} \).

### Step 2: Determine \( b \)
Next, we need to determine \( b \). From the graph, it appears that the function is increasing exponentially, meaning \( b > 1 \). However, we don't have another exact point to directly calculate \( b \), but we can hypothesize that the base could be \( b = e \), which is a common base for exponential functions.

So, the equation of the exponential function is:

\[
f(x) = \sqrt{2} \cdot e^x
\]

This is a reasonable assumption based on the graph's shape and the common use of \( e \) in exponential growth problems.

Let me know if you'd like further details or clarifications!

### Related questions:
1. How can we verify if \( b = e \) is the correct base for this function?
2. What would the equation look like if the base \( b \) were not \( e \)?
3. How does changing the base \( b \) affect the steepness of the graph?
4. What is the domain and range of this exponential function?
5. How would the graph change if we multiplied \( e^x \) by a constant other than \( \sqrt{2} \)?

### Tip:
When analyzing exponential functions, look for points where \( x = 0 \) to quickly determine the initial value, and use other points to find the base.

Find an equation for the exponential function graphed.

Learn how to find the equation of an exponential function from a graph. This problem demonstrates the use of the general exponential function formula f(x) = ab^x, and the use of e as the base for exponential growth. The graph passes through the point (0, sqrt(2)), making it suitable for students in Grades 10-12.

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