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.