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determine the equation of this function with base 3

To determine the equation of the function shown in the graph with a base of 3, we need to identify the function's form and specific characteristics, such as any translations or reflections.

Given that the graph appears to represent an exponential function, the general form of the equation is:

\[
y = a \cdot 3^{bx + c} + d
\]

Here, \(a\), \(b\), \(c\), and \(d\) are constants that affect the shape and position of the graph.

### Step 1: Determine the basic form
From the shape of the graph, it seems to be an increasing exponential function, which suggests \(a > 0\) and \(b > 0\). Since the function is based on 3, we start with \(y = 3^x\) as the base function.

### Step 2: Identify shifts and translations
1. **Horizontal Translation**: The function appears to have been shifted to the right, indicating a horizontal translation. Normally, \(y = 3^x\) passes through the point \((0, 1)\). Here, the curve appears to cross the x-axis closer to \((2, 0)\), suggesting the function is shifted to the right by 2 units. So, we have \(y = 3^{(x - 2)}\).

2. **Vertical Shift**: There doesn't appear to be any vertical shift since the asymptote is still at \(y = 0\). Thus, \(d = 0\).

### Step 3: Adjust the function based on transformations
Based on the observations, the equation of the function can be expressed as:

\[
y = 3^{x - 2}
\]

### Verification
Let's verify this by checking a couple of points:
- At \(x = 2\), \(y = 3^{2 - 2} = 3^0 = 1\), which seems to match the graph.
- As \(x\) increases beyond 2, \(y\) increases exponentially, which is consistent with the graph.

Thus, the equation of the function is:

\[
\boxed{y = 3^{x - 2}}
\]

Would you like more details or have any questions?

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Here are 8 related questions for further practice:

1. How would the equation change if the graph were shifted 3 units up?
2. What would the equation be if the base was 2 instead of 3?
3. How would you find the equation if the graph reflected over the x-axis?
4. What is the domain and range of the function \(y = 3^{x - 2}\)?
5. How would the equation change if the graph were stretched vertically by a factor of 2?
6. What point does the graph pass through if the equation was \(y = 3^{x + 1}\)?
7. How does changing the base of the exponential function affect the graph?
8. How would you determine the inverse of the function \(y = 3^{x - 2}\)?

**Tip**: Understanding the transformations of exponential functions helps in quickly identifying the function's equation based on its graph.

Learn how to determine the equation of an exponential function with base 3 using transformations and graphical analysis. This problem covers identifying shifts, translations, and the effect of base changes in exponential functions. Suitable for high school students.

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