We are given a graph of an exponential function of the form $y = a + 2^{x+b}$. We need to find the equation of the graph based on the given points on the graph, which are $(0, -1)$, $(1, 1)$, and $(2, 5)$. We need to find the values of $a$ and $b$.

1. Problem Description

We are given a graph of an exponential function of the form $y = a + 2^{x+b}$. We need to find the equation of the graph based on the given points on the graph, which are $(0, -1)$, $(1, 1)$, and $(2, 5)$. We need to find the values of $a$ and $b$.

2. Solution Steps

Let's use the points $(0, -1)$ and $(1, 1)$ to find the values of $a$ and $b$.

Plugging $(0, -1)$ into the equation gives:

$-1 = a + 2^{0+b}$

$-1 = a + 2^b$

$a = -1 - 2^b$ (1)

Plugging $(1, 1)$ into the equation gives:

$1 = a + 2^{1+b}$ (2)

Substitute (1) into (2):

$1 = (-1 - 2^b) + 2^{1+b}$

$1 = -1 - 2^b + 2 \cdot 2^b$

$2 = 2^b$

$2^1 = 2^b$

$b = 1$

Substitute $b = 1$ into (1):

$a = -1 - 2^1$

$a = -1 - 2$

$a = -3$

So the equation is $y = -3 + 2^{x+1}$.

Let's check with the point $(2, 5)$:

$y = -3 + 2^{2+1}$

$y = -3 + 2^3$

$y = -3 + 8$

$y = 5$

This confirms that the equation is correct.

3. Final Answer

The equation of the graph is $y = -3 + 2^{x+1}$.

Therefore, the answer is C. $y = -3 + 2^{x+1}$.