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What is the equation of the exponential function shown on the graph

To determine the equation of the exponential function from the given graph, we can follow similar steps as before. The function appears to be of the form:

\[
y = a \cdot b^{x + c} + d
\]

Where:
- \(a\) is a vertical stretch factor.
- \(b\) is the base of the exponential function.
- \(c\) is the horizontal shift.
- \(d\) is the vertical shift.

### Step 1: Identify the base and form
The graph is typical of an exponential function with a positive base greater than 1 (since the function is increasing). Given that it's an exponential function, we assume \(b > 1\). Let's assume the base \(b = 2\), which is a common base for such problems unless otherwise specified.

### Step 2: Identify transformations
1. **Horizontal Translation**: The graph crosses the y-axis at \( (0, 1) \), suggesting no horizontal shift, so \(c = 0\).
   
2. **Vertical Shift**: The graph seems to have an asymptote at \(y = 0\), indicating no vertical shift, so \(d = 0\).

### Step 3: Determine the equation
Assuming no vertical stretch or compression, \(a = 1\), and the equation would be:

\[
y = 2^x
\]

However, the graph appears to have a horizontal shift to the left, indicating the function might be of the form \(y = 2^{x + 2}\).

### Verification
Let's verify this by checking the graph:

- At \(x = -2\), \(y = 2^{(-2 + 2)} = 2^0 = 1\), which matches the graph's y-intercept.

This confirms that the equation is:

\[
\boxed{y = 2^{x + 2}}
\]

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

**Tip**: Always check points on the graph, especially intercepts, to determine specific translations and shifts in the function.

Learn how to determine the equation of an exponential function from a graph. Step-by-step solution covering identification of base, horizontal and vertical shifts, and verification of the equation. Suitable for high school students studying exponential functions.

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