Question

Based on the family the graph below belongs to which equation could represent the graph?

Answer 1

Option 1 is the correct answer.

Explanation:

As per the curve given in the question, we can infer the following:

1. Horizontal Asymptote (value of
   `y` where
   `x \rightarrow \infty`): as per graph horizontal asymptote is at
   `y = 3`.
2. Vertical Asymptote (value of
   `x` where
   `y \rightarrow \infty`): as per graph vertical asymptote is at
   `x = -2`.
3. 
   `y-` intercept (value of
   `y` at which
   `x = 0`): as per graph,
   `y`-intercept is
   `3.5`.
4. 
   `x-` intercept (value of
   `x` at which
   `y = 0`): as per graph,
   `x`-intercept is
   `-2.33`.
5. The values of
   `x \text{ and } y` can be positive and negative, both.

Now, looking at the options given in the question figure:

Option 2 is exponential function i.e.
`2^(x)` which can never have negative value of
`y` so option 2 is not correct.

Option 3 has
`y`-intercept as 3. So option 3 is not correct.

Option 4 is logarithm function, so value of
`x\ in\ log x` can never be negative. So, option 4 is not correct.

When we see option 1, all the 5 conditions mentioned above (as interpreted from the question figure) are satisfied.

So, the equation of the figure is
`y = (1)/(x+2) + 3` i.e. option A is correct.